Older ones do, newer ones have a diagnostic port you plug an adapter into for use.

You don't need to use it unless you are rewriting modules or performing certain diagnostic tests because the center display has access to all the information you would need via a diagnostic mode.

The data available is extensive, you can look at fault codes, and perform standard maintenance procedures like coolant bleeding. Listing what data is available would take entirely too long because it is literally everything the car monitors.

Supposedly the diagnostic mode has more features if you are in a service center, but I can't confirm that.

https://www.tesla.com/ownersmanual/modely/en_cn/GUID-B27A666D-866B-4766-B690-CCD1B66E2134.html

Possibly? There's conflicting info, it seems some model do, some don't and even models that are supposed to have it might not 🤔

edit: Forgot to mention, Tesla constantly updating hardware like it was software will end up biting owners in the ass as things break and there's no easy way to know what their car needs compared to what looks like the same car built 6 months later. There's a reason why legacy manufacturers work in generations.

OBDII hasn't been used for emission equipment exclusively from the beginning, it's the USB of the car world, so it's used for pulling info from all sensors and from the different computers and can be used to reprogram the car, so legacy manufacturers for sure use it in their EVs...

My 2013 does. Not sure what info it gives as I haven’t used it.

I used to work at a place that made smart chargers for EVs. They did all sorts of intelligent scheduling, V2H and V2G, grid response and load shedding, some really clever stuff. The standard for most charger interfaces allows for the vehicle to communicate a load of information to the charger, and almost none of them implemented any more than the bare minimum. I'm many cases the charger can't even tell how full the car's battery is, it just has to charge until the car disconnects itself and stops charging, and assume it's done so because it's full. So, I wouldn't be surprised if Teslas don't communicate as much over OBD as you'd expect given the standard it supposedly implements. Manufacturers seem to be quite content to keep that stuff proprietary wherever they can.