1

Anywhere really, central Rome is relatively walkable (depending on how far you are able to walk). As an example, I stayed at the Alessandro Palace near Rome Termini a few years ago and distance wise that was fine with me.

Refer to http://www.hostels.com for a list of all available hostels including reviews.

2

As above...the centre of Rome is not enormous.There are also buses,the metro etc.

Most people either stay near Termini (good for prices and transport links),Trastevere (good for nightlife,restaurants) or right in the centre.....good for walking around and everything else really,but usually the most expensive.

3

All the areas are safe,if you take the usual precautions.

Termini (like most train station areas) has some dodgy characters hanging around though ;-)

4

the centre of Rome is not enormous

Well, it sure isn't tiny, either.

If we can roughly agree on some rough boundaries--Trastevere to the west, Termini station to the east, Piazza Di Spagna (or, arguably, Piazza di Popolo) to the north, the Colosseum to the south, and the Pantheon basically the bullseye...I wouldn't want to walk from one end to the other--+if+ I had other walking to do in-between (and I absolutely would have other walking to do!).

It's all safe, it's all walking distance to the things you want, though if you're too far in one direction, you may want to take public transit to something on the other far end. Heck, at the end of a long walking day, I've taken a bus from Piazza Navona to Trastevere. Livin' large, I know...

Between hotels, B&Bs and apartment rentals, there are certainly hundreds, possibly a thousand, of accommodation options within that area. Most will be at least adequate. The more centrally located you are (Pantheon), the pricier it's likely to be. Happy hunting.