> Can a N100-class minipc” be installed inside of a wall with a touchscreen and serve as a PoE powered Home Assistant interface?

Yes. Generally only requiring a $10 PoE splitter like this: https://www.ebay.com/itm/134500605396

Some N100 class machines draw more power, but many don't, and there are more capable PoE splitters for a few dollars extra.

Use a USB touchscreen.