Do I have to install every nightly on LineageOS? Do I have to download and install every single nightly that comes up?

Are the later updates dependent on the earlier ones?
Every update would fully overwrite /system and carry all history changes, therefore just installing the latest build would sync you up to progress.

There will be (rare) occasions where the developer pushed patches that negatively affects the build (e.g. wrong CPU governor causing the phone to lag or heat up). On top of that, Android security patches, which are arguably the most important change for users, are only updated once per month. In light of these, although installing every nightly when it comes out is possible, I suggest updating monthly, plus making a nandroid backup and inspecting user feedback (e.g. respective device discussion thread on XDA-Developers) before flashing any build.