Yes, you can stain it on sawhorses. If you cover them with rags, then it will only be a few minutes before the stain is dry enough that you can flip the door over and do the other side. Apply the stain with an appropriate brush, and then wipe it with a lint-free rag before it tacks up. If the door is pine or another soft wood, use a wood conditioner before applying the stain.
For a door that doesn't see the elements, you can use your preference of exterior grade clear coat. Brushing is fine. Just don't put it on so heavy that it runs. That problem will mostly be alleviated by doing it on sawhorses.
For a door that doesn't see the elements, you can use your preference of exterior grade clear coat. Brushing is fine. Just don't put it on so heavy that it runs. That problem will mostly be alleviated by doing it on sawhorses.