Wow... that's a tall order, but I'll begin with a little bit of information which may or may not be of some use to you. I'm sure lots of others will chip in over the next few days. Close cover and shorter shooting distances usually call for smaller magnification. a 2-7x32 or so would be a reasonable call for up to about 200 yards. In more open country, a good all around choice is a standard 3-9x40. With the shorter distances, you coud go to something smaller, or even a fixed power, say 4x, rather than a variable. Super cheap scopes usually turn out to be regrettable purchases in the long run. Many guys get good service out of scopes in $150 -$250 range, while others swear by more expensive stuff. The mid-range Bushnells are pretty common ... Trophy, Legend (no longer available new), and Elites. Burris, Vortex, Weaver, and a few others can be had in a similar price range, and all have their fans.
Most mount set-ups include bases, which are attached to the rifle, and rings which are attached to the bases. The "rail" that you mention is likely a one piece base which fits across the opening on the action. A ring is attached to the front and back ends of the base. Some folks prefer this set up, believing that it provides extra strength and better alignment of the rings. Others prefer not to have that "rail" across the open action, and like the separate front and rear base set up. Both one piece and two piece bases are particular to the make and model of the firearm. Some manufacturers, like Talley, market units that have the base and the bottom part of the ring as a single unit, one for the front and one for the rear, which attach directly to the firearm. It's a nice set up and worth considering if available for your rifle.
Regarding the crooked crosshairs on your Bushnell... if they are still staight lines, and still cross at 90 degrees, the scope could just be canted to one side or the other. This is a matter easily remedied by loosening or removing part of the rings, and turning the scope slightly so the crosshairs appear perfectly vertical and horizontal, then re-tightening the rings. Weaver rings in particular can be a real PITA when trying to mount a scope with the crosshairs perfectly vertical and horizontal.
I'm assuming that your BSA is already drilled and tapped for bases, and guessing that Weaver likely makes bases that will fit your BSA ...should be available on-line or through any decent gun shop. You would then have your choice of fair number of makes and styles of rings to fit those bases. Most manufacturers produce rings designed to fit on Weaver or "Weaver type" bases. Weaver rings are relatively inexpensive, and will do the job, but there are others on the the market that I think most guys prefer ... Burris, Leupold, just to name a couple. Also, most offer rings in high, medium, and low heights. I like to keep my scopes close to bore (low rings), but some actions require medium or high rings to allow the bolt handle to clear the scope rear "bell" when the action is opened. The size of the front bell/lens can also come into play when determining whether low, med., or high rings are needed. Some guys prefer medium or high rings as a matter of personal fit on particular rifles.
