1. Balliol has a gym (next to the bar, ironically for rowers) on-site, which is a little old and not too pretty, but has a good range of weights, lots of rowing machines, bikes, weightlifting machines (sorry, not sure what they're all called) etc., but no elliptical trainers/running machines. Also, if you manage to get inducted, you can get access to the gym at Holywell Manor, where the graduate students live, which is apparently much nicer and does have a running machine (but only one rowing machine).
2. In first year, you don't get to choose your own room - you are asked to choose a 'band' indicating how much you are willing to pay for a room and en-suite rooms are the most expensive (on the whole) so choosing a higher price band increases your likelihood of being allocated one. Last year, a fresher was allocated an en-suite room but this depends on a) whether or not any such rooms are available after third and fourth-years choose theirs, and b) luck
In second year, most students choose either to live out in rented houses with friends, or to live in a flat Jowett Walk, which is college-owned accommodation (about a four-minute walk down the road from Balliol). All except a handful of rooms in Jowett are en-suite.
In third year, students are put at the 'top' of the housing ballot (people's positions on the list are randomly assigned) and get to choose rooms in college (or choose to live in Jowett/live out) in that order. There are about 40 en-suite rooms (or more that I haven't heard of?) in college and it isn't unheard of for people very low down on the ballot choosing these rooms. It just depends on how many people above you in the room ballot were willing to pay the extra rent of an en-suite room or even wanted one, each year.
3. kitchen facility for self catering
Some rooms have access to a kitchen (certainly all the main fresher staircases have kitchenettes, as well as a couple of others), but many people find it most convenient to eat in hall, certainly for dinner, and balliol has a student-run pantry which serves breakfast, lunch and dinner most days. All rooms on Jowett walk have access to a kitchen (there's one per flat), and obviously if you choose to rent a house outside of college in second year, you'll be hard-pressed to find one without a kitchen! Ours had two.
4. I don't think the Maths department is too far away - it's a short cycle or a slightly longer walk (10 minutes? I'm not sure, but you can probably search for it on google maps) up Banbury road from Balliol. What is further away is Ewert House, where all the Mathmeticians and some other science students have their exams in first and second year (not sure about third). This can be upto a 40/45 minute walk straight up Banbury Road from college, although there is a frequent bus service and taking a bike also significantly reduces this time. Anyway, there are few colleges from which you can easily get to Ewert house, certainly none in the centre of Oxford. Balliol has plenty of bike racks both inside and just outside college, as well as a bike-sale either in college or somewhere in Oxford (the union?) usually in first term, so most people choose to get a bike at some point.
5. Balliol doesn't provide fridges in your room (there are fridges in the kitchenettes, however), although you are allowed to bring a mini-fridge (up to four feet tall, or something, I'm not sure but one of the porters comes round to check your fridge is the right size eventually). These are usually large enough to keep enough food for one person in, though.
Hope that helped!