a) 1.20 can be patched to 1.23 easily by using the registered version patch. I'm not sure about the patch you speak of, but you can download one that works for sure from my server here.

b) 1.23 levels work in 1.24 fine. you'll even be able to play in 1.23 servers by doing some minior modifications to 1.24. On the other hand you cannot always successfully join 1.24 servers with 1.23.

c) Download section lists a number of fine downloads. If you're looking for multiplayer levels, I suggest you try out the latest anniversary bash collections. Those normally contain a lot of good multiplayer levels handpicked by the community.

Have fun.