The vanity couter top we purchased has a backsplash glued to it. The
backsplash is not flush with the back of the counter top, so if the
counter top is touching the wall, there is a 1/4" gap between the wall
& the backsplash. Is this standard, and if so, why? Is the gap there
to allow the bottom of a mirror to fit between the wall and backsplash?
If we don't fit the mirror behind the backsplash, should we just caulk
it or fill the gap with a strip of wood? Or detach the backsplash from
the counter top and re-glue it?

Most mirrors are attached with an aluminum strip at the bottom that
cradles the bottom of the mirror and picks up the weight. Try to get
the screws (long) into studs. Then, there are a few plastic clips
(again, try to get them into something solid) to secure the upper part
and sides of the mirror. You do NOT want a mirror coming off the wall!

As to the gap, simply caulk it. Something with a little "give" to it,
like latex or silicon caulk will do the trick. Pick a color to suit, or
one that can be painted. Do this before you horse the mirror into place.