1.) Buy T-money card in any subway station location, they have on screen touch shit so you can pick English and figure out how from there. Basically you're putting in ur korean cash into the card, so you can use it for subways/taxi/busses just by tapping it.
2.) i'd say restaurant food is cheaper than street food nowadays, especially late at night the owners will try to invite you into their restuarants and offer you free soup and shit. Meal is like $6-12 depending on what you order, and street food stuff are usually $1.5 - 4. Would you pay ~$8 and get a full meal, or 2-3 snack-like foods off the streets?
3.)$50 a night for hotel seems pretty cheap, sounds pretty low end tho if it's that cheap. Try airbnb, probably would find better place there. Keep in mind, spaces in korea is hella small so most places will feel crammed.
4.) i'd stay near insadong/seoul station area, it's pretty much the middle and can get anywhere within same amount of time. But since u gotta stay near gangnam gu, i guess it doesn't matter. The subway line will take you pretty much anywhere np.
5.) no fucking idea, didn't see that when i went this June
6.) Visa works for taxi and malls etc., but it wont work EVERYWHERE like streets stands, rando restaurants, basically places you think they won't have a card machine. Just go to any Korean bank, and exchange some dough to their currency so you have some pocket cash for places that dont take visa.
7.) idk.. porkcutlets, friend chickens, black bean noodles, seafood pancake, beef soups, bibimbap, seafoods, kbbq meats? but beef meats are pricey af, stick to pork/chicken if you gotta stay in budget lol