Hi Pavel, here are the differences as best as I can explain them:
1) "Participate" means to actively engage in something, whereas "take part" can mean the same thing or can mean simply engage without as much dedication/commitment.
2) Like #1, this one is about depth. If we're "checking" something, we can simply give it a quick inspection; however, if we're "checking out" something, we might inspect it with more depth. "Check out" can also mean borrowing books from the library.
3) "Currently" means it's happening right this moment. "Recently" means it happened in the immediate past. "Recently" could be happening right now as well, but "currently" definitely is.
4) "Assume" means to draw conclusions without proof. "Suppose" can mean the same thing as assume, but it can also mean to consider something for the sake of an argument.
I hope my explanations helped. If not, please let me know and I'll try to explain them in another way. If someone else can explain better than I can, please do so.