Is #2 visually graded, the most common lumber found in lumber yards ?

Typically, yes. I request kiln dried lumber when available, but it's rare
to find kiln dried around here for anything larger than a 2x8 or so.

Which would be better, 2x10 on 12 inch centers
or 2x12 on 16 inch centers?

Taller joists usually result in a stronger floor.

What happens if you use a 2X12 beam,
but put them on 24 inch centers instead of 16 inch?

A much shorter span rating, and you have to use thicker plywood to span
the 24" joist spacing.

From all the input received here, I now have no doubt that I can span
20 feet easily, using #2 southern pine 2 X 12 beams , on
16 inch centers.

I'd have doubts. http://www.cwc.ca/design/tools/calcs/SpanCalc_2002/
Choose US species and the loads you indicated - you can't span 20'
easily or even barely. Your house, your call.

A 20' wide building with a 2x6 supporting wall at each end would result
in a 19'-1" free span. That leaves several options:

Assuming a 30 Live/10 Dead (sleeping areas/attics)

2x10 at 12" OC = 19'-10" span

2x12 at 16" OC = 21'-1" span

Bump the load rating to 40 Live/10 Dead (all rooms)

2x12 at 12" OC = 21'-9" span

On the other hand, if the attic isn't tall enough to walk through and
we're talking limited light storage, you could size the "floor" as
CEILING joists.

Assuming a 20 Live/10 Dead (Drywall, No rooms, Limited storage)

2x10 at 16" OC = 20'-9" span

2x10 at 12" OC = 23'-11" span

2x12 at 16" OC = 24'-4" span


Obviously, it depends on what James plans to store up there, but I'd
probably opt for 2x12's at 16" OC.

Anthony