I am planning a similar vanity/backsplash situation, with tile on the floor. If the vanity is teetering, suggest shimming under it in a way you can't see, or install the shims and then add shoe moulding to conceal the shims.
Vanity should definitely be fastened to the wall. For the backsplash, my plan is to install tile (likely mosaic) on the wall, and caulk the vanity countertop/backsplash joint as an expansion joint. That allows for a little bit of growth/contraction between the countertop and tile without dislodging the tile. Hope this helps. Srry can't help with the sealer question.
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Will
central Indiana
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