It’s going to have some metadata to that effect yes, like a file index or number of parts or total extracted file size. I don’t know the details, I’ve used them I haven’t read the spec. rar is Rarlab’s proprietary format so there might not even be a public spec.
They’re normally all the same size except for the last part, so it’s not that file 1 is just an index.

Yes, it asks for the next part if it’s not in the same folder with the same name, doesn’t really make a difference what it’s stored on. Multipart zip and tar also exist.

Multipart archives still exist. They’re now used for file sharing websites that have a maximum file size. Before that they were for unreliable p2p networks, so you didn’t lose the parts you’d already downloaded when your peer goes offline. Originally it was to fit something big on multiple cd-roms or floppies.

Opening somthing.rar also reads the data in somthing.r01 through somthing.r15 etc