Yes. It makes sense but perhaps the meaning would be clearer if you wrote it as "I will make a random list of matches." or "I will make a randomized list of matches." Of course, you could also uses the verbs "construct" and "create" instead of "make" too.

(Assuming that I am holding a competition and will randomly choose people to be matched with one another.) If I understood your question correctly, then yes, that would be correct.

No. You must move the word "random" to say "I will make a list of random matches". If not, it sounds like your "making" is random, like you made the list accidentally.