Yes, backsplash should run past the cabinet as you don't want the incomplete look of the tile running short.

To the end of the wall. Now is the time to decide how you will "finish" the edges (specialty tile or metal strip).

Do I have to finish it or can I just keep the pattern going to the end?

Only the front face of tile (the part you see) is "finished", the edges are "raw". You will need a metal strip along the exposed top edge and end. (No metal strip is needed under the cabinets because the edge is hidden.) Lots of threads on Houzz about how to finish tile edge.

The metal trim is a Schluter edge

https://www.schluter.com/schluter-us/en_US/Profiles/For-Walls/c/P-FW

any competent installer will recommend that to you.

Instead of stopping on the cabinet or wall, and pending on the tile size you chose, maybe stagger the tiles like steps going down? or a rough edge (pending on the tile material)

Unless your kitchen design is modern and very streamlined, then end the tile at the wall corner.