Edited on 2014-12-09 0:36:16
Edited on 2014-12-09 0:31:02

As far as rust there is always the cosmetic stuff, the things you can see, but not structural. Bubbles around the park lights at the top of the front fenders, bubbles at the bottom valance somewhat hidden by the front pumper. This area can have allot of bondo here beginning on the front valance and up into the fender well openings.
Grab the rear bumper firmly with your hand and make a serious effort to move the car up and down. There should be zero movement where the bumper is bolted onto the body. Rust can be bad here, but impossible to see behind the bumper.
Shine a light in the fender well openings and check the floor areas, this place can be rotted pretty bad and someone may have hidden it with roofing tar.
Make sure the jacking points are still there _ this can't be hidden and will indicate just how bad the rest of the car can be.
Structural stuff is where the pan-hard rod mounts to the body on the inner rear right fenderwell. The pan-hard rod keeps the rear axel centered in the body. With out the pan-hard rod mounted to the body, the axle will be allowed to move from right to left. The fender skirts will be damaged from the tires rubbing up against them in cases like this
Stick a flash light in this area and make sure that the pan-hard rod is still mounted to the car.
Remove the rear seat, very bad examples show rot and rust bubbles where the rear leaf spring pockets come in contact with the seat well in the body. I've seen holes rusted right through showing the leaf springs _ don't buy a car like that.
If you find a car with perfect rear fender skirts, then be suspicious _ assume the car has had them replaced. Very few people will have looked after the car to the point where there is no rust on the fender skirts.
The usual stuff _ the inner rocker panels (use a mirror and a flash light), but if the jacking points are gone and the rocker panel is in good shape, then suspect that something has been covered up in this area.
The whole things about a rust search is common sense more then anything.

I can't advise you specifically about the auto trans, but older cars from the sixties have torque converters that do not lock and rob you of power and gas millage.
The shift linkage can give problems as they become dirty and are very often out of adjustment.

When it comes to the engine _ an engine is an engine... If it was badly rebuilt and the cylinder walls were over-honed to make the bores parallel again _ this is bad. The engine will not burn oil by getting by the rings, but the pressure in the combustion chamber will be forced into the oil pan area _ this is called blow-by. The outlet for this is at the top of the head _ at the front. This is routed into the bottom of the air cleaner so it will be burned when sucked back into the engine. If the car has excessive blow-by, the air cleaner are will have this excessive oily film all over it once the paper element has been removed.
The engine will be very noisy at cold start up _ when the bores have been over honed the pistons become too small for the bore and are allowed to slap around inside the bores. This is called piston slap. In not-so-bad cases this will become quiet when the engine warms up. Worst cases will always be very noisy on acceleration and deceleration.

Take the car on a test drive and have someone follow you. Drive the car down a long hill (preferably on a highway at day time)Have the person in the other car watch the exhaust just as the car starts the descent down the hill. Blue smoke out of the exhaust indicates worn valve guides when you take you foot off the gas while driving down hill.
The vacuum of the engine sucks oil past the worn guides and has maximum vacuum at this point because the butterfly valves in the carburetors are fully closed.

When the car is first started, rev the engine to at least 2500 RPM, but do not hold it there. A tight engine will yield no blue smoke and sound quiet and when I say quiet , there will be no noticeable knocking sounds.

After the car is fully warmed up, rev the engine to 2500 rpm again, a healthy tight engine will have the same sound when it's cold.

If you don't know what to listen for have an experienced mechanic listen for you _ or even better, some one who rebuilds engines for living.

My engine has been rebuilt (properly) the oil pressure is at 100PSI (bourbon tube gauge) when cold and settles down to about 50 to 70 PSI.
This is very high even for a new engine and I think someone at some point put in a high pressure release valve in the oil pump circuit.
An engine with worn main and connecting rod bearings will have low oil pressure and that pressure on a worn engine will be very little on the gauge.

A sloppy front end indicates the usual _ worn tie rid ends, steering box, steering linkage etc.
If it has power steering and the main shaft that comes out of the bottom of the box is dripping oil (they all do, but it shouldn't be excessive) Then the whole box is probably worn and the parts to rebuild them are not being made anymore (the last time I checked) XKS.com makes a kit to replace the old power steering boxes with a rack and pinion unit.

If the brakes have never been rebuilt, they will need it, especially if the car has been sitting in a car port or in an unheated garage. The bores in the brake calipers like to rust and leak.
The brake servo's wear out to the point that the brake fluid gets sucked into the engine _ huge clouds if gray smoke will occur and the car will use excessive amounts of brake fluid, but there will be no leaks because it's being burned in the engine.
If the flexible brake lines have not been replaced, then they will need to be, they either get hard as rocks or the inside gets soft and collapses _ and in this case the fluid can't flow properly causing uneven pressure in the system. The car will pull one way or the other or if the rear flexible line is collapsed then the rears won't get fluid. In this case the rears won't come on as they should and in a panic stop the fronts will have to much power _ this is dangerous.