First of all, welcome to Using English! I'm glad you like it here.
The first thing to note is that the choice of conditional statement does not necessarily indicate the intentions of the speaker. Thus in these two statements –
1. If you do X, I'll do Y.
2. If you were to do X, I would do Y.
– it is not possible to ascertain whether the speaker does indeed intend to do Y, if the condition is fulfilled.
It would be truer to say that #2 presents the same condition as #1 in a more tentative or deferential or remote way.
It's even more difficult to ascertain the intentions of the original speaker in reported speech, because the apparatus of reporting (backshifting, etc.) can make the form of the original text uncertain.
In your sentences:
1. He said he would do what we asked him if we fulfil some conditions.
— this could report either a type 1 or a type 2 conditional. Note that some (but not all) speakers would regard the if-clause as incorrect, and require this version:
1a. He said he would do what we asked him if we fulfilled some conditions.
2. He said he would have done it if we fulfil some conditions
— this is not quite consistent: the "he would have done it" refers to an unreal result in the past, while "if we fulfil some conditions" refers to a condition that is still open.
2a. He said he would have done it if we had fulfilled some conditions
— this conveys the meaning of a willingness to do X in the past, if we had done Y in the past. It reports a type 3 conditional.
3. He said he was willing to do it if we fulfil some conditions.
— this is fine: it reports "I am willing to do it if you fulfil some conditions". However, again, some speakers might be more comfortable with:
3a. He said he was willing to do it if we fulfilled some conditions.
Other members may disagree, though, as conditional statements are often much disputed on forums such as this. I may well have overlooked some aspect of the question. So watch this space!
All the best,
Student or Learner