Four nights eh? That doesn't give you much to work with. A quick Google search tells me that you guys are gonna be staying either at the one on 53rd and Lexington or the one on 49th between Lex an Park. There's a ton of great bars around that area either way. I would say anything above 30 dollars per meal would be kinda pricey especially with tip but at least to me. Be sure to carry your Passports on you at all times in case you guys waltz into bars at night.
Since you mentioned loving steak,
Peter Luger instantly popped into my head out in Brooklyn. I've never been there before but it's one of the most famous and best steak houses in the city.
As for Mediterranean food, your best bet for that would be to go out to the Astoria section of Queens since that neighborhood houses a large Greek community. However, given the short amount of time you're gonna be here, I'm not sure if it would be worth the trek.
Bareburger might another place worth checking out if you want to have exotic meat burgers such as Ostrich and Boar (don't get the Boar if you guys go; it's not very good IMO). There's one 32nd and 3rd Ave. and you guys can always hit up Third and Long across the street from it afterwards.
I'll give a you quick list of bars you can hit that in the Hell's Kitchen/Midtown East area based on having been there during GAF meet ups or otherwise:
Valhalla (on the corner of 54th and 10th)-large beer selection, great atmosphere, good pub food. We've hit up that place a couple of times during past GAF meet ups and I've been there on a few dates. Girls seem to love it. It can get loud though so be warned.
Pony Bar (45th and 10)-Another GAF meet up favorite with a great beer selection (get the Pear cider and a grilled cheese sandwich; they're favorites in that place) and cool atmosphere. Warning: it's a bit small as it can get crowded if you go during peak/happy hour (17:00-22:00 in EU time).
Barcade (various locations)- There's one on 24th between 7th and 6th (AKA Avenue of the Americas but only tourists call it that) and another one on St. Marks Place; both are in Manhattan. There's also another one out in Wiliamsburg (Brooklyn) on the L train. I suppose it's worth checking out at least once.
The one on 24th has retro arcades from the 80s whereas the one on St. Marks has more 90s arcades and fighting games such as
Street Fighter III. Personally I like the latter location the best. Oh and if you do use the bathroom at the latter location I would recommend using the leftmost one; I left a message on the wall with my GAF name. It's still there to this day.
I think that's enough for now since I can go on and on. The Lower East Side is chock full of bars and restaurants that you hop around. For any other recommendations, feel free to browse through this thread or check out the (rather outdated) OP.
It's pretty great. Love buying prints there.