The mathematically minimum possible solution is:
Credit to supercat and user1540815! I overlooked an important fact in my first draft.
If the trolls stand in one line, there are 4 possible identities for the first troll, 3 for the second, 2 for the last. Overall, this is 24 possible combinations.
Trolls: W (truth),X (liar),Y (random),Z (mute)
Any question to Zrowag (mute troll) will give you only one piece of information: he is the deaf one. This will reduce your pool to 6 combinations for the other 3 trolls.
Each further question is a strict yes/no question, without any tricks. This will get you at most 1 bit of information. So whatever clever question you design, the next question will at most half it to 3 combinations. Next question will half it to 2 combinations and the fourth and last question will decide it.
How does it work if I ask a Troll first who isn't Z ? I get more than 1 bit of information for any question to a new troll before I know who Z is. If I get an answer, I will also know that The troll I asked is not Zrowag and can at best half the remaining possible combinations.
So if I ask the first troll and he does answer, I will reduce the possible combinations from 24 to 18 (because there are 6 combinations where Zrowag is the first troll) and then can half the remaining space to 9.
The second question needs to be asked to a different troll, so I will either find Zrowag to reduce the combination-space to at most 3 possible combinations ( since with a good question my 9 combinations left after the first question will be 3 for each possible position of Z)
If the second Troll also is not Zrowag, my second question will again eliminate 3 combinations (because I know Zrowag is not on place 2) leaving 6 combinations, which my yes/no question can again split in half to 3 possible combinations.
If I get lucky I now only need one more question, since my 3 combinations will be two combinations with Zrowag in Place 3 and one with Zrowag in place 4, So if I ask the troll in place 3 and he is Zrowag I'm done, otherwise my question can split it again to the last remaining option and done.
So if I get lucky I can get it in 3 Questions, but worst-case still remains 4 Questions.
Example Questions to get it in Four:
I will arbitrarily number the 4 Troll 1-4 for the Place they are standing in (assume they stand in a line)
First I will try to find a Troll who is either W or X (either tells the Truth or lies)
Question 1: Ask Troll 1:
From the trolls who can speak, is the Troll closer to you more likely to lie?
I ask W (truth teller). He has to decide which one of X(liar) and Y(random) is more likely to lie. He will say "yes" if X is closer to him and "no" if Y is closer. If he answers "yes", I will ask the closest troll to him next. If he answers no I will ask the furthest troll next. - So I will ask Z or X next.
I ask X(liar). If W(truth) is closer to him, he will lie and say "yes". If Y(random) is closer he will say "no". By choosing the same as in 1a I will ask Z or W next.
I ask Y(random). He will randomly say "yes" or "no" so I will choose W,X or Z next.
I ask Z(mute). He will not respond, I know who Z is and ask the same question to Troll 2. This will cost me one question, but I know who Z is and will continue and need 3 more questions overall.
Question 2: Ask Troll 2 or 4 (where we know he is NOT Y(random))
Can you hear me?
Scenario 2a: Yes! He is W(truth).
Scenario 2b: No! He is X(liar)
Scenario 2c: He remains silent. He is Z(mute) - I will ask the same Question to the next troll in line and will have used one Question for Zrowag.
Question 3: Ask the Troll again (I know he is W or X so I know if he will tell the truth)
Is this Yijlob? (pointing to the first Troll who spoke)
Scenario 3a: He confirms it (Either W saying yes or X saying no) I know who Yijlob is. The remaining two must be the opposite of the current troll and Zrowag
Scenario 3b: He denies it. (Either W saying no or X saying yes) I know the first one is now the opposite of my current Troll (either W or X) and the remaining two others must be Yijlob and Zrowag
If I already know who Zrowag is, I am done now (and used 4 Questions)
Question 4: Ask the Troll again if you have not found Zrowag yet
Is this Zrowag? (pointing to one of the two remaining Trolls)
I now have sorted out the last two troll and am done.
Daredevils may only need 3 questions
Since our goal is only to know who each troll is, the riddle doesn't mention we need to survive with this knowledge for long. Using the option of asking a question which a troll cannot answer, we have a third outcome to a question: The troll may kill us (even if he does not, X and W can only lie and tell the truth, so are not able to answer certain questions)
Since we may die, we can only use this on the last question. Since we can theoretically always get down to 3 possible permutations with 2 question, the third question could divide them into yes/no/die and in the worst case we would know the solution when the troll kills us.
I would love if someone finds actual questions to solve this in three, although it is probably stretching the rules of the riddle. - the proposed solution for 4 questions is not optimal and will leave us with 10 permutations after the first questions and get only to 4 after the second question. But it could be possible with better questions.