Basement – Is it safe to block the hole in the wood in the basement

That is odd. If that's how the mice got in then you want to fill the blocks with gravel & then smooth that over with concrete or cement. If the mice didn't enter there then "Big Gap" spray can foam would be the insulation choice & best treatment.

If you'd like to use the block voids to secure the sill of the stud wall(s) then you can screw cleats to the sill plate bottom. The cleats would be set into your concrete, cement or foam flush to the top of the blocks & the sill would be screwed to them later.

Preferably, the cleats would just be PT2x2 chunks that only snug against the basement side of the block void to hold the sill against the wall or sill with spacer blocks behind & at the wall...if the plan is to align the sill & studs to the face of the blocks & just run drywall over the blocks for a fully flat wall.

pros - looks better, smoother surface that makes less dust

cons - seals the surface up well which prevents moisture from evaporating from the block. if its 100% sealed on the outside, it wont be an issue, but if you have any moisture, it will trap it in the block.

painting block is an extremely common commercial/industrial finish process. the trick is to use the right approach and material.

step 1 - fill and seal with concrete prep coat. this material is usually sprayed or put on with a texture gun, but if you want you could probably do it by roller (i wouldnt bother, just spray it)

https://www.dulux.ca/pro/products/masonry-coatings/perma-crete-ltc-concrete-block-and-masonry-surface

step 2 - paint over with whatever you want

omit the sealer and you can count on 10 coats minimum to seal it up properly.