Steph - sorry to read of your loss.
The legal ownership of the registered and unregistered garage/property passes to Dad following her death.
The TIC, wills and trust created relate to their beneficial ownerships
1. No as the legal ownership has not been transferred.
2. If a voluntary application was submitted it would be in his name. You don't register what might happen next.
3. You don't have to advise us that the garage and drive are now in his sole name. The death is sadly factual so can be proven as and when the actual legal ownership is transferred in the future
But you can update the register -
HMLR Guide Page · HM Land Registry