I can help a little with Seoul, although it has been 12 years now since I was last there. First of all, just know that when you fly in, you're not going to land in Seoul - you'll be about a 45 minute drive from the city. Count on a taxi ride or a train ride into town. The train is cheaper, but the taxi is much easier with luggage.
Be aware that unless things have changed, people smoke in public places much more than we do. You may be hard pressed to find a restaurant without a cloud of smoke, and when you book your hotel, if you are not smokers yourselves, be sure that you ask for a non-smoking room.
Good food is cheap and plentiful. Bulgogi, kalbi, tofu kimchi chige, bibimbap... mmmm. Sanitation, however, may not be up to the standards you're used to, especially if you eat street food. I've seen the same chopsticks used without washing at street stalls. I got around this by keeping my own in my purse.
I can't give you a hotel recommendation simply because when I was there, we were military, so we usually stayed on base when we were in town.
I can tell you this - Seoul is immense, and it doesn't really have a city center. The best advice I can give you is head to the library and check out a stack of guide books (I like Lonely Planet). See what two or three activities or sights interest you the most, and stay close to them. There are fabulous museums and historical sites in the city, and some of those are clustered in the same area.
One of the most fun things I ever did while in Seoul was to attend a show of traditional Korean music and dance.
Even the locals don't drink tap water.
Public bathrooms are not necessarily segregated by gender. If you see toilet paper in the wastebasket, that means you shouldn't flush any, either - in some areas the plumbing is quite old and temperamental, and you don't want to cause an overflow.