I don't see why not. SD cards are nothing but a storage media, like an SSD, HDD, USB drive, etc. In fact, that's how you typically run an operating system from a Raspberry Pi and similar hardware.
Although you may want to make sure your SD card is fast enough and can sustain frequent reads and writes. Not that it wouldn't be able to do so but they've been known to wear down much quicker than traditional hard drives.
Sure. Raspberry Pi devices do it all the time using either the SD card or a USB thumb drive. Not a great way to run a complete operating system but it can be done.