In time, the customers will get to know your dh. They will know the work he does and his attitude towards customers.
Once he has built a solid relationship with the existing customers, then taking over (and changing business names) will hopefully run smoothly with little hassle.
It is a good idea to let existing customers know that it is now under new management and that the new owner is a current employee.
Astrid made a good point about limiting the customer database. People may be frightened off if too many changes are made when/if you takeover. A lot of people don't like change. It's best to do these things slowly. See what sort of business the classic cars brings in before making any decisions.

Get the figures thoroughly checked out by a good accountant. The Accountant will also tell you the best way to establish the business, ie partnership, trust etc. And they will also value the business for you.

As for your last question, what does he really want to do? If he is really passionate about taking over this company, then there is no harm (if you can manage it financially) in your dh staying there and getting a better idea of the work they do, and getting to know the customers.
If he's really not sure what he wants to do, and wants to wait a little before buying a business, then there's nothing stopping him from leaving and finding something else that pays more money.
From the sounds of it though, this is something you both want to do. If it were me, I'd encourage dh to stay where he is. Like I said above, get to know the clients and the business a little better.