Whether it is or is not, is determined by the source from which it is received. The goddess of comfort and cookies would likely include anesthesia. The bloody god of war I think would expect someone to suck it up and drive on with the mission.
Not in 5E, no. As far as 5E is concerned, it is literally impossible for anyone under any circumstances to ever suffer such a degree of injury that it cannot be completely remedied through natural healing over a course of eight hours sleeping. If we dismiss the assertion that everyone in the world has a Wolverine-style healing factor, the only alternative is to accept that real damage just never occurs. (Short of death, of course, which is the only damage that can't be healed overnight.) (Or extreme exhaustion which, while falling short of death, can potentially take longer to recover from than any amount of HP damage.)