If the mural is not permanent, you can use plywood as a backer. Better yet, use wallboard. Cement board will work, too. I like the idea of the recess, and the velcro idea is smart. Use regular tile mastic to stick the tiles, and sanded grout. I am assuming you have experience with tile murals, so won't bore you with details on how-to. You will still have to consider how to finish the edges between the mural and the recessed tiles - I would recommend tubabd tile grout in a color that closely matches the grout color. It can be easily scraped off when the time comes to replace, and it will prevent moisture from getting behind the works and mucking things up in the future. Good luck, and would love to see pics of the progress as well as the finished product.

Could be Acrylic panels like 3D-Form etc. There are companies that can add any image to the acrylic. I would also inquire about wet wallpaper. Many of those have beautiful patterns such as the ones you posted. https://www.wallanddeco.com/en/18-collections--Wet-System

There is more reason for a backsplash than just to make it look good. As the name implies, a backsplash is there to prevent things from splashing onto the painted wall. So Please finish off the walls with a backsplash that coordinates with the tile mural that is over the cooktop.

At the end of the day, you can use whatever you want as long as it doesn't void the warranty on the glass in the door.