I have a Maine Coon that I got from a shelter and she's HUGE and my place is TINY. She was sick when I got her, I knew she was sick, it's the reason I permitted myself to get her, because I thought well, at least what I can offer is better than dying in a cage. We're very close because I had to feed her with an eyedropper for a little over a week. That's not the way I recommend going about it -- but I so agree with Halla that a cat that has been bonded, raised, nurtured makes a good cat. The younger you meet them, the better. You can train cats with food -- it's easier than any other way. You should handle them a lot when they're little, and interact with them as often as possible. People think cats sleep all day because they like to sleep, but often it's because they're bored. Give them at least 10 minutes of attention a day. I brush my cat every night and then play with her for a few minutes and she's happy. Also if you can, do things on schedule with them, e.g., I brush the cat every night at 10 pm. She knows she will have that at least if nothing else. She also knows when it's 10 pm. Keeping a schedule with at least one thing helps them feel secure.
I have never had a cat with a litter box problem -- for me, it's been set the box up, bring the cat home, put the cat in the box, the cat goes to the box when it needs to, no training involved. As long as you keep the box clean, they will use the box (normally usually generally).
Vasilisa's right about premium food. If you feed premium food, there's usually not as much shedding.
Long hairs are a lot of trouble to clean up after, and harder to bathe. Sometimes I've had to take my cat to the groomer and it's $70 a pop. Short haired cats, you can run a damp warm washcloth over them once a day and be done with it. So I recommend short haired cats. I'm allergic too and some days depending on what's in the air outside as well, my system really struggles, antihistamines aren't good for you long term, ditto decongestants. Siamese are really smart and they are good companions, but they really are mouthy, and if they don't get enough attention, they will start destroying your stuff.
Look for a short hair that takes to you -- let the cat pick you. That's the best way.