The terms and conditions of the Tesla warranty would dictate whether or not the warranty would still apply in the event of the vehicle being written off, so you would need to check this in the vehicle documentation, but most warranties require the vehicle to be serviced in accordance with the maintenance schedule at a main dealer, and if substantial repair work has been carried out by an independent garage then it is likely that the warranty would no longer be honoured.

As for the Tesla app and software updates, these should continue to work as long as the vehicle is fully functioning and is registered with the Tesla network, but there may be an issue if the fact that the car was critically damaged was automatically uploaded to the Tesla network. As for using the Supercharger network, Teslas ordered after January 2017 have to pay to use this network - whether your car will be able to use them will again depend on the vehicle's status on the Tesla network.