Build a cmu block wall on two sides, face it with stone and do a concrete counter top acid stained and sealed and cantilevered 12" so you can pull bar stools up to it

I agree with the other comments, wait until you white wash, but I would be tempted to paint some colour of the walls if you cannot remove the trim (it might be there to tidy up the joins).

Skirting gives it a nice streamlined look with the rest of the house. Quad just looks like you couldn't be bothered finishing the job. I live in a house where that happened before I moved in and I hated it for many years! Skirting board gets my vote.

Your installation is correct. All homes I have had with skirting and tiles, never had caulking between skirting and tile. Only between top of skirting and wall. There should only be a 1mm to 2mm gap under skirting. Be careful not to mop with excess water. I have always used a steam mop.