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Don't confuse your two separate problems. (And I'll bet they are not related.)
If the rudder shakes there can be several reasons. On some boats, the design is simply poor and the prop is too close to the rudder. If someone replaced the rudder with one that extends too far forward, that could cause the problem. Sometimes the wrong prop (too big) or a too-long prop shaft has been used. Sometimes, a boat is simply designed with the engine and prop shoehorned in afterwards and it is a poor design so the prop vibrates the rudder. Sometimes the prop shaft support has come loose from the hull, which can mean you're about to flood and sink. Sometimes the cutless bearing (the wear tube that the prop shaft passes through in that strut) is simply worn.
All you can do is inspect for visible wear or motion, you should not be able to "shake" the shaft or support at all, they should be rigid. When you do this the engine should be OFF and the key should be IN YOUR POCKET or otherwise secured, so there is no chance of being hit by a moving prop.
Last of all it could be an unbalanced prop (damaged or worn) but the only way to tell that, aside from visible damage, is because there's no other reason for a shaft shaking so you take the prop off and have it checked for unbalance. If you run down the list one at a time, you'll find the answer.
You really need another owner (you may be able to find an owners' group online) to tell you if the rudder shaking is normal for that boat.
On the overheating...that can be so many things. Anyone who calls himself a mechanic, who can't find out why an engine is overheating, should go back to being a carpenter. Exhaust problems, fuel problems, timing problems, cooling system problems....it is a long list but a thorough tune-up and checkout can find many of them. After that, the cooling system can also be checked and the only "holy mystery" left is a blocking in the exhaust elbow, which should be checked and cleaned or replaced every decade or so anyway.
Yes, a bad prop (perhaps wrong size/pitch) might cause everything, so the starting point might be to dive with a light and try to get the specs from the prop, they should be stamped in the body of it someplace. If not, then at least measure the diameter across the blades, and try to find out if that is normal for your boat.
I find the hardest part of freediving on a boat is not smashing my head against the hull while I'm floating back up, so if you have anything to protect your head, even a wool watch cap, that really helps.