1. Is there an advantage to adding copyright info to photos in the camera vs. doing it in LR?
2. How do you add copyright info to thousands of photos at the same time?
The basic metadata is stored in exactly the same place, so in that sense it doesn't matter - and people should do both.
The only advantage in setting the in-camera menu is that the images on the card are tagged, so if for some reason you just send the card to a client (or someone borrows it...) you don't have orphan works floating about.
There are many advantages to applying metadata afterwards, such as:
In LR (and all other Adobe applications) setting metadata in bulk is easy - just build a metadata template and apply it to all your selected images. In LR you can do this during import, from the Library module, or during export. The only caveat is that metadata templates have no internal logic, so for example you can't make them look up the copyright year from the capture time.