Most calibers use the same bolt carrier, just a different bolt.
9mm uses a mag well adapter either welded to each mag or temporarily inserted into the mag well. You will probably change the hammer and bolt too. 5.56 hammer is to short.
If you shoot a lot of .22 in a m16, clean the gun very well when you are done. There have been some kabooms linked to 22 gunk in the 16.
7.62x39 can be finicky on feeding due to the shape of mags. google m16 frankenmags and c-products. Some folks are lucky with them, some aren't.
"9mm has a very long history of being a pointy little bullet moving quickly" --Sevens