I have seen either way; stopped right at the bottom of the cabinet or to the end of countertop, it is whatever your prefer.

The backsplash is part of the counters, so go to end of them. You can use a thinner piece of the mosaic as your vertical edge, if you're concerned on the end.

Yes, stop even with the cabinet edge.

Also, I'd change out the white switches and outlets for a color that blends with the splash or ss if you are using it elsewhere.

Considering that your counter extends further out than your cabinets, why not stair step the product from top to bottom? Would give an interesting visual affect.

Keep going 4" lower so you won't need the lip.