Fellow cyclist here
Manchester is
pretty flat pancake flat, so you really shouldn't have any difficulty physically speaking.
1. Nope, Manchester has (contrary to popular belief) lower rainfall than the UK average. There's the potential for the odd bit of drizzle, but tbh if you have a decent coat you'll be absolutely fine. I have a pair of waterproof trousers somewhere (I didn't buy them specifically for cycling) but I can't remember the last time I wore them whilst on a bike.
2. It's a bit fast, and not really an ideal introduction to cycling, though I've cycled it myself. This is not the Netherlands! They're currently about to
shut Oxford Road to private vehicles, so I think my preference would be to go up Anson Road for a short distance, then left onto Oxford Place, then turn right and go onto the segregated cycle path, then continue onto Oxford Road, and then go down Charles Street onto North Campus (have a look at a map, it'll make more sense). It's not that much further, and it avoids most of the traffic.
3. It may not snow - we're in the middle of a big city, and Manchester's snow tends not to pitch even when it does in all the surrounding areas. Personally, I've never given up cycling in the snow. However, if it does snow then you can always consider getting the bus for a few days if you don't feel safe. If you're from a country where it snows every year, be ready to laugh (or cry) at how little snow it takes for most of Britain to grind to a halt (schools shut, trains delayed, the works). It's about two inches. There is, however, some good advice on cycling in the snow here
http://www.theguardian.com/environment/blog/2010/jan/06/tips-cycling-in-snow4. There's a bike shed at Dalton Ellis, so put it in that and you should be absolutely fine. However, I will strongly recommend that you use a
D-Lock regardless of where you've parked it.
5. You'll want to wear a coat and gloves in the winter, but you're not going to die!
You don't say if you're coming to uni with a bike, but if not then Bike Doctor (a lovely little workers co-op that won't rip you off on Dickenson Road), Edinburgh Bike Co-Op (definitely the posher end of the market, located on the corner of Wilmslow Road and Oxford Place) and Bike Boutique (rather like Bike Doctor, located next to the Sugden Centre) are all good places to buy one / get repairs done. Don't, whatever you do, buy a bike from Halfords. They've got a dreadful reputation amongst cyclists.