1a) No hurry: their rights remain after you buy, so be nice to both of you and get the purchase out of the way.
1b) Open Rent is fine - you don’t transfer the tenancy, you create a new one, but only have the start date after you have confirmed that you have bought the place! Yes, you can start the process before the place is yours, but be safe and not have the tenancy signed by you until after you know you own the place.
2a) Because of the above, if the tenancy started some years ago it might not be an AST that they have. As you said, they are a sitting tenant. In this case they can always say no to any change, even if they say yes now, so be wary.
2b) If you don’t want a life long tenant, be safe by asking the landlord NOW for a copy of the existing tenancy to check what you are letting yourself into before you buy, just in case the tenant does not go ahead with the proposed AST that they have agreed to.
2c) Check that there are no more than two tenants, otherwise you may be inheriting an unlicensed HMO that may need renovating to comply with local authority or government rules for HMOs.
2d) Valid certificates must be available and issued before or with the proposed AST, so check Gas is less than one year old, EPC less than five years old, and electrical is less than the renewal date stated on it, which can be less than the maximum of five years, and any rectification work stated on it has been completed within 28 days of its issue.
2e) Provide the government current Rent Guide before they sign the agreement.
3) Check with Land Registry that the seller is in fact the owner and no debts have been placed against the property - the conveyancer should check this - otherwise they will become your debts if not cleared before contract exchange!
4) I have found that being a DIY landlord is extremely time consuming by having to keep up with legal changes, and it is difficult to manage a difficult tenant from a distance, so consider using an agent to help you, remembering that you, and not they, have sole responsibility for legal compliance. Both the local authority and your insurer may insist on six monthly inspections for example, so make sure they take place, and have fire alarms installed on each floor.
5) Arrange for an emergency call-out person to be named in the agreement.
6) Don’t forget to register / protect the deposit if you have one, but it is not a legal requirement to have one. If one was provided before, the current landlord should arrange for this to be available or returned to the tenant.