Mount your side splashes as normal. Move the vanity flush with one of the sidesplashes. Get a another length of sidesplash and mount it horizontally like a little raised shelf you can put your toothpaste, cologne, liquid soap bottle, etc out of the way... like it was intentional so you can clean the vanity top without having to move everything. The gap below the shelf will be visible if the face of the vanity is a solid cabinet. If it's a vanity with legs like a table then the gap won't really stand out.

For the gap against the back wall and the vanity will the the hardest to make look good.

Get some project board and cut the width of the gap and length of the front side of the cabinet. Paint it the color of the cabinet or wall color.

Dry fit your ptrap before deciding which side to put the extra sidesplash piece, might be easier to connect with the vanity shifted right or left. If you have a medicine cabinet on one side, put the extra piece on that side.

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Nah. Visualized it and looks weird, both with sidesplash mounted to wall or mounted to flush with vanity.

Vanities and vanity tops can be sold separate. Hold out for finding a top that'll fit that 63" or have a custom counter company cut you one to size and back and side backsplashes along with cutting holes for faucet and drop in sink. Browse their remnants... a jumbo slab measures 63" on its side.