For such a small distance 1.5 x 2.5 will be plenty big enough. You dont
need any intermediate uprights, but adding one in the 50cm wide wall
will make everything more solid.

Wood finish: anything, makes no difference.
Noggins: anything, I'd just use a bit more 1.5x2.5

(You can even do a job like this with scrap wood and oddments)

Note that anything weighty that goes on the wall will need a noggin in
the wall to support it, and you cant add them later, so plan. Dont
forget to run a wire if you want a wall light.

Final note: always use 12mm PB. Its just =A31 more than 9.5 and much
stronger and less prone to getting holed later.

NT