Would be great to get the box menu only if you stop the car in a garage.
You stop there, then the box menu opens automatically.

And if you had a crash on track so that you can not manually drive in the garage, then it should be a time penalty of maybe 10 minutes or something if you have to press Esc to open the box menu.

So if there is a qualifying for only 15 minutes. and you crash in minute six or did flip on the top. then the qualy would be over for you

Hard but realistic and cool in my opinion

Can we explain the carrying on in quali by cadging a scooter ride back to the pits and getting in the T car?

Yes of course, but like i wrote, there will be a time penalty. Maybe there could be a counter in the box menu runnning down (i.e. 10min). and only then you can join.

So if there was less time left in qualifying, then there would be no chance to continue qualy

Nee... you could just spectate and then go an die box--why bother?