The rumbling doesn't come back immediately after (assuming I wait about 2 - 5 seconds) even if at the same speed, or within a few mph of it. 

It doesn't consistently happen at any particular range of speeds, it's mostly random, maybe happening a bit more at the 35 - 45 mph range, but that was the average speed I was going, so it'd make sense to fall in that range most often. 

It physically shakes the car, somewhat noticeably. 

If I don't let up on the gas, it goes away within a few seconds, without much noticeable change in noise or volume, although I only did this once or twice, because it seemed like a bad idea. 

I have yet to see if there is any difference between overdrive and drive

Almost every part is original, with the exception of the radio, alternator, radiator and battery. 

PS: 
It was bought from a private party. 

No idea if it's has a transmission fluid change, probably not, should I change it? 
What other fluids would be good to change, and which ones not good to change? 

It doesn't seem to like starting so much, it almost never starts on the first try, but will consistently start on the second try. Why is this? Does it say anything about the health of the car? What can I do?