"Can anyone recommend some varieties?"
Any variety you have a taste for. Apple trees can be excessively pruned to be kept at the height you want. Apple trees marked "standard","dwarf", or "semi-dwarf", has to do with putting one type of apple (e.g. Honeycrisp) on different rootstock (i.e. grafted into the roots of a different apple species) to keep the tree naturally smaller (based on the natural size of the rootstock).
A "standard" tree can still be pruned into whatever size you want - some people prune them and train them along wires like grape vines or up the sides of walls like climbing roses (google 'espalier' for cool pictures). Not all trees can be pruned dramatically like that - e.g. nut trees can't be - but apples can and are the most common example.
So, yea, get whatever varieties you personally enjoy!
Buying store-bought apples, my go-to apples were always "Honeycrisp" as an eating apple, and "Grannysmith" as a slightly-sour baking apple. When I planted my trees, I planted those, but I also planted multiple other varieties to see what else I might enjoy. There may be better-tasting apples that stores won't carry because they select breeds for long shelf-life or uniform appearances.
My trees are just starting to reach fruiting age, so I haven't gotten a chance to see what my varieties taste like, but am looking forward to it.
"Will I need to cross-pollinate?"
Yes. All this really means is, have two or three compatible apples within 30 ft of each other, and you'll be fine.
"compatibility" mostly is just making sure they have overlapping blossoming periods, so pollen from one can reach another.
Just figure out what varieties you want, then look up if they are compatible with each other, or what other varieties people recommend pairing with them.
Sounds complicated, but it's really not. Just pick what apple varieties you want, post them here, and ask people if they'll cross-pollinate well, or if there is a substitution or addition they'd recommend.
But if you arbitrarily picked based on what your tastes are, there's a decent chance they'll happen to be compatible anyway.
"If growing in containers, is it realistic to expect any fruit?"
Yes. I've never grown them in containers personally, but heavily pruned apple trees do still produce apples.
Get a decent size pot to give room for root growth. Also you'll have to pay more attention to watering compared to planting in the ground; you'll also need to eventually work more nutrients into the soil of the pot after a few years, as the soil will be stripped of all beneficial nutrients by the tree and will need more.
Personally, I'd just plant them in the ground, even if you want them small, and keep them pruned to the size you want. Your USDA Zone 6B (same zone as me) can support apples outdoors just fine.
"Do apples suffer any insect, fungus, or similar problems I need to watch out for?"
Depends on your area. In Missouri, my apples were eaten by bugs that lay eggs in the inside of the apple shortly after blossoming, and then when the apple is almost ready for harvest, they burrow their way out and eat the apple.
Since I don't use pesticides, the supposed solution to that that I'm doing this year, is covering each individual apple with a little baggy once blossoming is finished (I'm using disposable ziplock bags, but in Japan they make little fabric baggies, and some people in the USA make bags out of nylon stockings or something).