Hello Yoda! Your questions cover a variety of forum categories here including "Transportation", "Food, Drink, & Entertainment", "Packing" and even "Favorite Places" but no questions, really, for the "Hostels & Other Lodgings" forum. Since I'm a bit confused as to what to do with your posting I'll leave it where it is.
First, have you conducted a SEARCH of
Hotel Wellington in this forum category? Possibly some of the previous reviews have comments on the area of the hotel. You'll find the hotel to be in a very nice area in the Salamanca District.
Salamanca District is considered "higher class" than other districts in Madrid and so people typically dress nicer. But if you go to a restaurant in the area I doubt you'd be turned away until it was a REALLY fancy, black-tie-type place and those are few.
Taxi transportation could cost anywhere from 22-30 Euros in my opinion.
Is there a lot to do in the area? No. Not realy, unless you like shopping. Remember, this is mainly an old, residential district with big houses and international embassies. The thing most worth seeing close to this hotel is the Retiro Park, the Puerta de Alcalá, and that's about it. All other things worth seeing are a bit further away requiring longer walks, bus, taxi, metro.
Regarding your non-meat-eating-classification, check the "Food, Drink, & Entertainment" forum category using keywords
MADRID SALAMANCA VEGETARIAN.
Good luck and enjoy your time in Madrid! The hotel is very nice!
Saludos, MadridMan