Project 64, pretty much.
There's some cool stuff going on with the MAME/MESS N64 driver in the accuracy (LLE) department, but not all available yet, and very slow. (and lest it sound like it, it isn't my work, but our old pal TheOtherMoogle's)
Hopefully you aren't referring to the current beta. The site's been getting some updates recently; eventually it'll surpass the other emulators. The biggest competitors right now are Mupen64 and 1964, but in some cases even Nemu64 does better. As of beta progress that has been dying out though. In fact it's kinda dead Nemu64, but I don't need to tell you all that.
"2) 1964 0.9.9 (never getting updates again?)"
1964 was never discontinued. It's most likely schibo and Rice are waiting for information--although that could be possible in multiple senses.
"3) Project 64k... or... I forget what it is, but the one that was modded with netplay."
1964, Nemu64, and Mupen64k, Mupen64++ also have NetPlay. Nemu64 was the first, but Project64 was faster. Therefore someone hacked version 1.4 to insert a NetPlay feature, but you get a lot of desynchronization.
Years later (and very recently) okaygo joins EmuTalk and releases a NetPlay emulator that beats both of these engines comparing Project64k as "NetPlay without sync." The reason it never got too popular is because most people were focused on playing Super Smash Bros. on Project64k and didn't check for updates.
1964 does alright with NetPlay--better than Project64k. 1964 is even less popular than Mupen64k. Later the project evolved into Mupen64++, but after college influences and loss of the sourcecode the project died. Now a group of developers are working on a Win32 port of the new "Mupen64Plus" project.