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I can do (and this is minimal):

3 questions (for both variations, this is minimal because there are only 3 possible responses and 3²<4!)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the following dictionary (see dictionary below, Batman will have to recite this but I won't add it here), and took the first/second/third change this for each question - each corresponding to the question number asked letter of the identifier, will either the letter be Y and you are telling the truth, the digit be N and you are lying or the letter be P and either you will lie and say yes or tell the truth and say no?

Then:

Yes means Y, no means N and pass means P since the option for P is a head-exploding statement. Then you can reconstruct the three letter identifier, back-index into dictionary and find the order!

Dictionary:

JPRT: YNY
JPTR: YNN
JRPT: YNP
JRTP: YPY
JTPR: YPN
JTRP: YPP
PJRT: NYY
PJTR: NYN
PRJT: NNY
PRTJ: PPP
PTJR: PNY
PTRJ: PNP
RJPT: NYP
RJTP: PYY
RPJT: NPY
RPTJ: NPP
RTJP: PPY
RTPJ: NNP
TJPR: PYN
TJRP: PYP
TPJR: NNN
TPRJ: YYP
TRJP: PNN
TRPJ: YYN

Examples:

Riddler, letter=Y: "The letter is Y, and I am telling the truth. So true, and I say yes."
Riddler, letter=2: "The letter is not Y. The letter is N, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, letter=P: "The letter is not Y or N. The letter is P, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, letter=Y: "The letter is Y, but I am lying. So false, but I say yes."
Penguin, letter=N: "The letter is not Y. The letter is N, and I am lying. So true, but I say no."
Penguin, letter=P: "The letter is not Y or N. The letter is P, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying. Note - the Joker will never be given a head-exploding question

I can do (and this is minimal):

3 questions (for both variations, this is minimal because there are only 3 possible responses and 3²<4!)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the following dictionary (see dictionary below, Batman will have to recite this but I won't add it here), and took the first/second/third change this for each question - each corresponding to the question number asked letter of the identifier, will either the letter be Y and you are telling the truth, the letter be N and you are lying, or the letter be P and either you will lie and say yes or tell the truth and say no?

Then:

Yes means Y, no means N and pass means P since the option for P is a head-exploding statement. Then you can reconstruct the three letter identifier, back-index into dictionary and find the order!

Dictionary:

JPRT: YNY
JPTR: YNN
JRPT: YNP
JRTP: YPY
JTPR: YPN
JTRP: YPP
PJRT: NYY
PJTR: NYN
PRJT: NNY
PRTJ: PPP
PTJR: PNY
PTRJ: PNP
RJPT: NYP
RJTP: PYY
RPJT: NPY
RPTJ: NPP
RTJP: PPY
RTPJ: NNP
TJPR: PYN
TJRP: PYP
TPJR: NNN
TPRJ: YYP
TRJP: PNN
TRPJ: YYN

Examples:

Riddler, letter=Y: "The letter is Y, and I am telling the truth. So true, and I say yes."
Riddler, letter=2: "The letter is not Y. The letter is N, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, letter=P: "The letter is not Y or N. The letter is P, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, letter=Y: "The letter is Y, but I am lying. So false, but I say yes."
Penguin, letter=N: "The letter is not Y. The letter is N, and I am lying. So true, but I say no."
Penguin, letter=P: "The letter is not Y or N. The letter is P, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying. Note - the Joker will never be given a head-exploding question

I can do (and this is minimal) [EDIT: this may be invalid because of undefined behaviour]:

3 questions (for both variations, this is minimal because there are only 3 responses and 3²<24)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the list of twenty-four permutations of those four letters in lexicographical order, and took the ones/threes/nines a different one of the three options for each question digit of that number in ternary, is the following statement true: 'either the digit is one and you are telling the truth, the digit is two and you are lying or the digit is zero and either you will lie and say yes or tell the truth and say no'?

Then:

Yes means 1, no means 2 and pass means 0 since the option for 0 is a head-exploding statement. Then you can reconstruct the number, back-index into the aforementioned list and find the order!

Index of order:

JPRT,JPTR,JRPT,JRTP,JTPR,JTRP,PJRT,PJTR,PRJT,PRTJ,PTJR,PTRJ,RJPT,RJTP,RPJT,RPTJ,RTJP,RTPJ,TJPR,TJRP,TPJR,TPRJ,TRJP,TRPJ

Examples:

Riddler, digit=1: "The digit is 1, and I am telling the truth. So true, and I say yes."
Riddler, digit=2: "The digit is not 1. The digit is 2, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, digit=0: "The digit is not 1 or 2. The digit is 0, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, digit=1: "The digit is 1, but I am lying. So false, but I say yes."
Penguin, digit=2: "The digit is not 1. The digit is 2, and I am lying. So true, but I say no."
Penguin, digit=0: "The digit is not 1 or 2. The digit is 0, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying.

I can do (and this is minimal):

3 questions (for both variations, this is minimal because there are only 3 possible responses and 3²<4!)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the following dictionary (see dictionary below, Batman will have to recite this but I won't add it here), and took the first/second/third change this for each question - each corresponding to the question number asked letter of the identifier, will either the letter be Y and you are telling the truth, the digit be N and you are lying or the letter be P and either you will lie and say yes or tell the truth and say no?

Then:

Yes means Y, no means N and pass means P since the option for P is a head-exploding statement. Then you can reconstruct the three letter identifier, back-index into dictionary and find the order!

Dictionary:

JPRT: YNY
JPTR: YNN
JRPT: YNP
JRTP: YPY
JTPR: YPN
JTRP: YPP
PJRT: NYY
PJTR: NYN
PRJT: NNY
PRTJ: PPP
PTJR: PNY
PTRJ: PNP
RJPT: NYP
RJTP: PYY
RPJT: NPY
RPTJ: NPP
RTJP: PPY
RTPJ: NNP
TJPR: PYN
TJRP: PYP
TPJR: NNN
TPRJ: YYP
TRJP: PNN
TRPJ: YYN

Examples:

Riddler, letter=Y: "The letter is Y, and I am telling the truth. So true, and I say yes."
Riddler, letter=2: "The letter is not Y. The letter is N, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, letter=P: "The letter is not Y or N. The letter is P, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, letter=Y: "The letter is Y, but I am lying. So false, but I say yes."
Penguin, letter=N: "The letter is not Y. The letter is N, and I am lying. So true, but I say no."
Penguin, letter=P: "The letter is not Y or N. The letter is P, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying. Note - the Joker will never be given a head-exploding question

I can do (and this is minimal):

3 questions (for both variations, this is minimal because there are only 3 responses and 3²<24)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the list of twenty-four permutations of those four letters in lexicographical order, and took the ones/threes/nines a different one of the three options for each question digit of that number in ternary, is the following statement true: 'either the digit is one and you are telling the truth, the digit is two and you are lying or the digit is zero and either you will lie and say yes or tell the truth and say no'?

Then:

Yes means 1, no means 2 and pass means 0 since the option for 0 is a head-exploding statement. Then you can reconstruct the number, back-index into the aforementioned list and find the order!

Index of order:

JPRT,JPTR,JRPT,JRTP,JTPR,JTRP,PJRT,PJTR,PRJT,PRTJ,PTJR,PTRJ,RJPT,RJTP,RPJT,RPTJ,RTJP,RTPJ,TJPR,TJRP,TPJR,TPRJ,TRJP,TRPJ

Examples:

Riddler, digit=1: "The digit is 1, and I am telling the truth. So true, and I say yes."
Riddler, digit=2: "The digit is not 1. The digit is 2, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, digit=0: "The digit is not 1 or 2. The digit is 0, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, digit=1: "The digit is 1, but I am lying. So false, but I say yes."
Penguin, digit=2: "The digit is not 1. The digit is 2, and I am lying. So true, but I say no."
Penguin, digit=0: "The digit is not 1 or 2. The digit is 0, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying.

I can do (and this is minimal) [EDIT: this may be invalid because of undefined behaviour]:

3 questions (for both variations, this is minimal because there are only 3 responses and 3²<24)

Ask the following questions to three people:

If I took the four letters J, P, R and T, put them in the order corresponding to the order you are in where J=Joker, P=Penguin, R=Riddler and T=Two-face, indexed it into the list of twenty-four permutations of those four letters in lexicographical order, and took the ones/threes/nines a different one of the three options for each question digit of that number in ternary, is the following statement true: 'either the digit is one and you are telling the truth, the digit is two and you are lying or the digit is zero and either you will lie and say yes or tell the truth and say no'?

Then:

Yes means 1, no means 2 and pass means 0 since the option for 0 is a head-exploding statement. Then you can reconstruct the number, back-index into the aforementioned list and find the order!

Index of order:

JPRT,JPTR,JRPT,JRTP,JTPR,JTRP,PJRT,PJTR,PRJT,PRTJ,PTJR,PTRJ,RJPT,RJTP,RPJT,RPTJ,RTJP,RTPJ,TJPR,TJRP,TPJR,TPRJ,TRJP,TRPJ

Examples:

Riddler, digit=1: "The digit is 1, and I am telling the truth. So true, and I say yes."
Riddler, digit=2: "The digit is not 1. The digit is 2, but I am telling the truth, so false. The digit is not 0. So false, and I say no."
Riddler, digit=0: "The digit is not 1 or 2. The digit is 0, and I am telling the truth. If I say no, then it's true, because I say no. If I say yes, then it's false, because I don't say no. So pass."
Penguin, digit=1: "The digit is 1, but I am lying. So false, but I say yes."
Penguin, digit=2: "The digit is not 1. The digit is 2, and I am lying. So true, but I say no."
Penguin, digit=0: "The digit is not 1 or 2. The digit is 0, and I am lying. If I say yes, then it's true, and I should have said no. If I say no, then it's false, and I should have said yes. So pass."
Two-face/Joker: refer to Riddler if telling the truth, Penguin if lying.