Oh weird, that's a new issue I haven't seen before! I don't have anything specifically wired for touch/gamepad but I assumed a touchpad would work by default. I can definitely see the right click being an issue though since you have to hold right-click while moving the mouse to rotate the object in your hands. I didn't think about how that would work on a trackpad.
Did you collect the first fruit hanging over the chair? If not, you'll need to collect it before you'll be able to buy any other furniture. You can't pickup/move anything in the environment once it's placed through by design.
Also, the music by a really talented artist called Abstraction! Their work is actually here on itch and a large chunk of it is available free for use in jams. I highly recommend their stuff!