Glasgow is not dangerous at all, nowhere near as dangerous as most large American cities. Its almost unheard of for a student to be violently attacked, and crimes like rape and armed robbery are pretty rare outside the suburbs. I have lived in Glasgow for 20 years nearly, and I have NEVER had any reason to go into the rough areas of Glasgow. The centre and west end are absolutely fine. Admittedly, its a little more lively and scary at night than in the day, especially Friday and Saturday, but its really nothing to worry about if you take some sensible precautions.
There are amazing museums in Glasgow (Kelvingrove museum and the Burrell are World Class), Kelvin Park is right next to Glasgow uni and is beautiful, and there are plenty of music and cultural festivals.
Edinburgh tends to be colder in Winter than Glasgow, and invariably gets more snow. It is a smaller city. And its also got barely any actual Scots in it, you might as well stay in America. Glasgow's West End, where the Uni is, is stunningly beautiful, even better than Edinburgh imo because its not got the same dark and depressing architecture (of course both cities have more than their fair share of modern dark and depressing architecture, but you don't need to see it very often).
Money wise, Glasgow is certainly cheaper. Edinburgh is about the same as New York in terms of consumer prices, though both have much lower rent than New York of course. Both cities are entirely walkable, you only need to catch the occasional bus, which is pretty cheap. Though Glasgow does have the added benefit of a Subway between the two most important streets, Byres road in the West End, and Buchanan Street in the centre. Edinburgh is a little more of a hike, harder to find your way around at first until you get to know it, and lots of hills.
Honestly, both are pretty equal in terms of all your criteria, and I'm probably being a little biased, but the fact is, Edinburgh is the more expensive city, so if you're avoiding Glasgow because of horror stories about how dangerous it is, or because you are drawn by the romance of Edinburgh and haven't looked into all the cultural and scenic areas in Glasgow properly, then please think again, because the reality is Glasgow is not the city it was 50 years ago, its a safe, fun, cosmopolitan, beautiful city.