When you have made your mind to buy a certain classic car and settle with the price, take a good look at the car before you hand over the money. Check all the details, especially the weak areas you already noted in your list. Check the engine, does it make noises? Is it overheating? Are there parts over worn? You can also bring a friend who knows a lot about cars with you.
When you are done with the car physically, move on to the paperwork of the car. If one or some parts of the cars have been professionally built in the past, the paperwork must have shown it since most owner keep all the receipts. Make sure that the car registration document is up to date and the numbers of the engine and chassis all match.
If the car passes all the checking, now you can hand out the money and bring the car home. Now that it is in your possession, treat it with utmost care and pay lots of attention to it. Have fun and drive safe!