I think both answers are right, depending of the context, for example, I think if you already mentioned those 2 guys you can write the text without 's And if you have not mentioned those 2 guys you must write theur names with 's This is only my point of view, because I think that is the best way to write the things clearly

Yes, but it needs to be clear whose approval it is. You can of course use 'their' as long as it is clear who "they" are.

It depends on whether you're refering to them as a collective (perhaps two team leaders), or as individuals. If a collective: We need Mike and Grace's approval (we need their collective approval). If individuals: We need (both) Mike's AND Grace's approval (emphasis on the and). A simpler example might be: Mike and Grace's marriage (their marriage) Mike's and Grace's opinions of marriage are different (Mike and Grace each have an opinion)

"Since we have Mike's and Grace's approval" - this is good :) They are two individuals, each offering approval. Two possessives :) If they are a unit and only have one opinion between them, they need relationship counselling :) But they would also need only the second apostrophe. See, if you like, https://proofreadmyessay.co.uk/writing-tips/apostrophes-joint-ownership/