Well, yes, there should be a simple contract stating what you are buying and how much is being paid for it. Otherwise, the seller could cash your cheque and then pretend it was a private loan, repayable in 10 years' time
Much and much has been written on this topic in the past but basically you do a very thorough inspection, with all inspection covers removed. Plugs out and cylinders borescoped, especially if there is the slightest possibility (which there will always be) of the plane having sat for ages at some stage (many for-sale have been, which is why they are for sale). Obviously all ADs checked both in logbooks and on the plane itself.
On the test flight, all avionics checked including all autopilot modes. What you are buying is a nontrivial machine.
How old is it, total time, equipment, etc? Email me if preferred. In fact I think you have my details but I can't tie up your nickname to your real name