This can be done, various phase detectors will handle the level issue differently.
Discriminator, as used in FM radio. However, a three-state phase/frequency detector with an appropriate loop filter will often do well enough without a bunch of fussy resonant circuits.
Yes, as is a double-balanced mixer.
There's no fundamental reason -- a downconverter that's going to a non-zero IF ("image reject") may need better phase matching, but it's probably mostly the manufacturers.
"Phase Lock Loop Circuit Design" by Dan Wolaver is a good book. Not a classic, but I think it could be if it caught on. It seems to be pretty clear for self study (but I wouldn't know -- I took the class from Dr. Wolaver).