The college library is LOVELY. As far as living facilities go, it can vary from staircase to staircase. Some have proper kitchens like you would have at home and corner baths whereas others don't. It depends which price bracket you apply for. Cooking facilities aren't really a big issue really though, as you can get meals in hall every meal time apart from Sunday lunchtime.
Most of the rooms are lovely, although none of the 1st years will get ensuite because you will be living in college. In subsequent years you will be living in more modern accommodation, slightly further away from college.
I wouldn't say that there is a stereotypical Tit Hall student exactly. I guess that typically you would know a large proportion of the other undergrads, be involved in a society/sport, will have had a go at rowing, enjoy your subject...
As for the SPSers, I'm not really sure. But if you are into debating you should join the Cambridge Union, both to watch debates and also get involved if you want to.