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Collections are a way for you to organize kata so that you can create your own training routines. Every collection you create is public and automatically sharable with other warriors. After you have added a few kata to a collection you and others can train on the kata contained within the collection.
Get started now by creating a new collection.
Actually I just re-checked the wiki, it seems that the katas do give you a set amount of score (which you can see at the bottom of the page I linked above) based on the kata difficulty "as well as an internal score that counts toward your next rank" (this part I don't really get), my bad/
https://github.com/Codewars/codewars.com/wiki/Honor-&-Ranks#leveling-your-rank :)
No, problems don't show how many points they get you on completion. You get more "score" (which is different from Honor) the more difficult the kata is, but it also depends on your kyu. F.e. you, as a rank 8kyu, would get a lot more score from completing a 6kyu kata than I, as a rank 4kyu, would.