Does the car work if the cellular connection is physically disabled?

Of course.

Yes, otherwise it would be unusable by anyone living in a city, given that underground parking (esp multi-level one) tends to have no reception at all. Not even mentioning driving through countryside stretches on road trips.

The only things that won't work with no cell connection while you are in the car are things like spotify app in the car (but you can still use spotify through the car bluetooth connected to your phone, if you preloaded the songs on your phone) or being able to see live stats of the car in your phone app.