You still take fall damage, any server, single player and not.

The description for the mod says there is a cave finder, but there's neither a config for it (description says there is one) and the default keybind I doesn't work either, so I'm asking if the features were removed or if they are just broken.

Description is outdated, yes. The render hack hasn't been in for several versions. But you do not specify a version of MC either.

Oh sorry, my bad on that one. I'm using the latest version on CurseForge for 1.12.2

https://minecraft.curseforge.com/projects/simply-hax/files/2444017

Re-downloaded and single player nofall works now, will test later on servers. Sorry for posting an issue without basic troubleshooting first.

Note the current implementation just tells the local player it has creative powers. It cannot override a remote server (you will be kicked when flying is disabled).