I use a portable butane stove like this. Butane is not the best option (they make a propane version) especially at altitude but I have used it at 8500' and it still works. I use old pots and pans that I have had laying around.
There are just so many meals to make! Either way, the list I gave you will make nearly everything. Anything else you might need, you should be able to find in normal grocery stores. Assuming you can get soy sauce. It is always helpful to buy a portable butane burner like this. They are cheap and easy to find. Perfect for hot pot on your dining room table.
Dude, you should get a propane hot plate so you can at least make some eggs or hotdogs and stuff like that.
you're not actually meant to cook the meat yourself. sure, you can fuss around a bit, but if the waitstaff sees you doing too much it's an immediate sign that they're not doing enough. if you really want full control of the grill, you might try Takashi. it's japanese barbecue owned by a korean chef that grew up in Japan.
the truth is Korean barbecue is very easy to do at home and way more cost efficient. you can probably get the grill at H Mart on 32nd. if not, it's cheap on amazon. the pork belly (ssamgyupsaal) served in restaurants usually isn't even marinaded or seasoned. all you need is to buy the pork (pre-sliced), some lettuce, an onion and some ssamjang sauce. the grill is good to have anyway.
that said, it's your first time having Korean barbecue and you're not likely about to go out and buy a butane gas burner. if you want to try a different restaurant, consider Madangsui or Shilla in K Town. they're both no-frills, but consistently good. i don't care for the hot newcomer - Kang Ho Dong Baekjeong. it's overpriced for small portions and limited side dishes. not unlike Chinatown, service in K Town generally sucks, especially if your party is all non-Korean.
These probably aren't BI4L, so flame away, but it's perfect for your situation:
http://www.amazon.com/Deluxe-Butane-Burner-Stove-Free/dp/B000BVC4NY
These are butane powered... You can find them local at restaurant supply places and oriental markets. They are very powerful but have good control to cook on. My office uses them in the cafe for making everything, and they look VERY well used. They are also apparently safe to use indoors without fancy ventilation, though I'm sure there are limits to far you'd want to take that.
Maybe not perfect for outdoor use since the flame isn't well protected, but it's cheap and will do the job you have at hand now very well.
I would buy local so you can test out the valve/igniter, the QC is iffy, some makes are better than others.