It depends on the company, but most do, with certain stipulations like not on your partner's business, not if they have bans etc. Usually they allow another driver that may not be your partner - eg if you are single you may need your dad to drive it occasionally etc. As long as all the relevant documentation is done.
Our company do but don't promote the fact in the hope you won't let them. I honestly belive that a partner should as the car is provided as a perk and you do get taxed so it should be able to be used as you would your own car maybe with the restriction of spouse/partner only plus any member of company.
Almost certainly yes, but when you're given a car (or opt to take one, as there could be a downside to this for you financially too) you should be given a copy of the rules, either in the form of a policy or perhaps a guidebook. This should also cover things like what to do if you have an accident and what to do about maintenance. Your new employer shouldn't mind you asking to see a copy of this now, which would help answer your question.