I don't sense a connotation of difficulty or effort in get + prep. Get gets used in many, many phrasal verbs, where its core meaning of "acquire" mostly gets out of the way and serves as a placeholder for something else. There is often a connotation of acquiring the result in some irregular, direct, crude way, though. For example, get out means to exit, but exit suggests leaving in an orderly manner, such as through a door, whereas get out just means to get yerself out of the location somehow or other. Get in means to enter, but you don't "enter trouble", you "get in trouble".
The word come usually holds onto its core meaning, so it often suggests motion, or orderly motion, more strongly than get. But there is no rule. Whenever someone tells you a rule like that, tell them to "get off their high horse". :) English phrasal verbs work mainly by extending and varying familiar precedents in higgledy-piggledy fashion, not by orderly rules.
Below is some messy native-speaker reporting on the kind of stuff that you can't find in dictionaries and that can only be learned from experience, not from explicit statements.
Abby had climbed a tree and couldn't get down.
Abby had climbed a tree and couldn't come down.
Come down has a slight suggestion of smoother, more orderly motion. Get down means to move yourself down by any means or in any manner at all.
Dad stopped the car and told me to get in.
Dad stopped the car and told me to come in.
You don't ordinarily "come in" a car (except, uh, in this sense). In the right context, it works, though, such as if you knocked on the door of the car; then the situation is similar to knocking on the door of a room, where the normal response is "Come in." Another situation is if the story is being told from the perspective of someone inside the car: "Dad opened the passenger door and Kinzle came in and joined us." I'm sure you could "go in" a car in the right context, too. Very formally, I suppose you could "enter" a car, but I've never heard that. But ordinarily, you "get in" a car. That's nearly always what one says (at least in the United States).
She forgot her keys and got into the house through the window.
She forgot her keys and came into the house through the window.
She forgot her keys and went into the house through the window.
These all mean pretty much the same thing. Got into has more of a connotation of doing things in an irregular, nonstandard way. Came into makes me imagine seeing her enter the house from a point of view inside the house. Went into makes me imagine seeing her enter the house from a point of view outside the house. Got into doesn't suggest any perspective.
The hill was so steep we had to get off and push our bikes.
The hill was so steep we had to come off and push our bikes.
I suppose it's possible to say come off your bike but that would be very unusual. It would suggest getting off your bike in order to come to some other point, suggested by context. For example, if I said "Come off your bike!" I would probably mean to get off your bike and come to me. Normally you just "get off" a bike. The only real synonym I can think of is "dismount", which is extremely formal.
By the way, because get off is so ambiguous (you could also "get off a hill"), you would normally word it this way:
The hill was so steep we had to get off our bikes and push.
A car stopped and two men got out.
A car stopped and two men came out.
These mean almost the same thing. There is indeed a slight connotation of orderliness in the came out sentence, which the got out sentence lacks. I also find myself imagining the men as seen from the outside of the car with came out—or rather, imagining it more clearly than with got out.
