Is it ok to use wood as a backsplash behind the bathroom sink k?

Only if the paint and top coat will be heat and fire resistant due to it being behind the stove.

While that will offer some moisture resistance, have you considered using wood-look tile? Tile not only works well in wet environments, but will be easier to clean if your backsplash gets hit with anything other than water (soap, shaving cream, hairspray, toothpaste, etc.).

I used wood wainscoting as my kitchen backsplash. I coated it with several coats of polyacrylic many years ago. Never have had a problem (stove not near backsplash.). So, I would say yes as long as you seal the wood real good.

you have to make sure it is sealed correctly

Think I would make it marine epoxy, holds up better to wet environment. Make sure to get the crystal clear, non-yellowing one.

no

if you use a sealant over wood, make sure you seal it on the wall side too otherwise you may install a hidden black mold farm ...

Be sure to seal ALL sides and ends of EACH piece of wood used! (Test a piece of wood over several days to be sure the 4 coats of poly are sufficient to protect the wood.) Because you will be using this in the bathroom as a backsplash, be sure to place the wood on backer board because of its proximity to water and moisture in the bathroom itself.

I don't see why not. You don't get the same kind of splatter that you would in the kitchen. Tooth paste would be your worst and it washes off.