This thread already has the answers to most of your questions.
The car is usable while it's downloading an update, but not when it's applying it. It even warns you that it will need 25 minutes. Assume it's 40min to be safe. Choose a time when you don't need it.
The car will notify you of an available update on LTE, you don't need to connect wifi for that. The actual download will not start until you connect wifi though.
EDIT: your sub issue is covered in another thread. It's fixed in some recent update, don't remember the number. Be patient, or don't use Sentry Mode in the meantime.