For my answer I'm considering the sides of the coin to either have a star or be blank for ease of writing. I also will write out the number for peoples name, and use the character for guesses to try and keep things legible. (Ex. Mr Three guessed 4). Also when typing out sequences, + means star and - means blank. (Ex ---- means no stars / all blanks)
I agree with Kate Gregory's Answer but there is a flaw in the logic.
in the second flip she said:
Mr Zero surrenders, which he would do if he had guessed (in order of the possible combos) 1, 0 (but he would have done so a turn earlier), 0 or 4, or 4. But for the last one, again, he would have done it a turn earlier. Therefore Mr Zero is 0 or 1, and he can't be 0 so he is 1 and the first two tosses are **. Also Mr Two is a 0.
however that's not true. because its also possible that...
Guessed 3 and the result was -- . It was noted that Mr Zero could have guessed 4 and conceded when +- happened. However this is ignored at the end and it's assumed that they only could have guessed 0 or 1. The assumption of the results being -- is correct but couldn't have been determined by the logic given.
How I got to my answer:
I listed all of the possible flip orders. I've lettered them for ease of reference. There are only 8 because we are told the result of the last flip.
After that I assigned guessed to people based on the clues.
From First Flip:
In A through D, Mr Two would had to have guessed 4, and in E through H they would have had to guess 0.
We have Mr Zero guessing 3 for A/B, 0 for C/D, 4 for E/F, and 1 for G/H.
I am ignoring Mr Three for now.
At this point we are told Mr One is still in it to win it. but with a single flip left, there would only be 2 possible outcomes left. so if Mr Four wins and Mr One is still in, Mr Three must be out. Which means in Alphabetical order Mr Three would have guessed: 2, 0, 3, 1, 3, 1, 4, 2
Now we set Mr Fours guess to whatever the right answer is and Mr One to one above that.
So the guesses would have to be (in name order)
however there are still 2 clues I've ignored up until this point.
We cannot have a set of guesses where only a single person guessed their name/number
which means that A, D, and E are all out.
leaving B, C, F, G, and H.
Then there is the last clue I've ignored.
Mr Three panicked when the second coin was flipped.
so lets look at our remaining options:
In B Three guessed 0 and I think it would have been weird for them to panic when 2 blanks come up in a row as that's exactly what needed to happen for them to even have a chance.
In C Three guessed 3 and the result was a second star. Which while it's not required that they get a star, it's still better than a blank. So I would find it weird for them to freak out.
In F they guessed 1 and the second flip had come up blank after a star which again they needed to happen for them to win. I know people who would still do this, which is why I didn't rule this out when first discussing flip 2.
In G we see the exact opposite of B, two stars in a row when they guessed 4, which is exactly what needs to happed.
Lastly there is H which had 2 Stars come up in a row. In this case Three guessed 2 and they actually had some leeway in this one. either result could eventually have won for them, but it's less likely if the result was a star. So for me this is the only one that makes sense for them to panic for.
so with that my guess is:
H) +++- , and the guesses in Name order are 1, 4, 0, 2, and 3 respectively with no one guessing their own name/number.
EDIT: Adjusted some formatted that I realized was off and fixed a spot where i excluded C for the wrong reason as was pointed out in the comments