9 sessions in 5 month is one session every other week in average. That's an awful lot of time to forget everything they've learned over and over again. I think your problem is specific not only to RPG but to didactics in general. So in addition to the cheat sheets I'd propose a sort of "bootcamp": A weekend with at least two almost day long sessions in a row so that there's something sticking before weeks long of forgetting.
First thing in this weekend would be creating the cheat sheets for each player/character. Ask the Battlemaster Fighter's player to support you in helping the others, because he seems organized enough and has his cards already.
Then run a demo campaign, which is designed to use the major points of each cheat sheet at least twice. Help them as much as needed on the first day. Practice wheels come off for second day, i.e. scale up difficulty as much as requiring them to use a bit more of their potential than they've been using before, let them fail (non-catastrophically), if needed.
Let me answer your explicit question "How do I, as a new DM myself, encourage or punish my players to do their homework on their characters abilities, features, spells and general game mechanics?" more explicitly:
Don't.
Let them learn by doing. That's way more fun and will help them ease the learning. Try to improve the conditions, so they will remember what they learned.