How do you use "go in", "come in", "get in", "go out", "come out", "get out"?
I think "go out" and "come out" mean the same thing and the only difference is where people are: in the same place or different places. Right?
For example, when I and my daughter go for a walk and we both are in the hall, I can say: open the door and go out (of the house). Not "come out". But if I am already outside (she is still in the house), I should say: come out. Not "go out".
The same with "go in" and "come in". If she is outside and I am inside the house: come in (the house). If we both are outside: go in.
What about "get in", "get out", "get off", "get on", "get down"? Is it rude to use them?
I know that "get in" and "get on" are used when we want to ask someone to enter a car or a bus/plane/train. What about lifts? Are "get in (the lift)", "get out (of the lift)", "get off (the lift)" rude?
"get on the chair" = "climb up the chair"?
"get down the chair" = "climb down the chair"?
Is it better to use "leave" and "enter" instead?
Thank you for your help!

Question

Answer

This is super context-specific. Your explanation regarding the general use of come vs go is correct, but I would not call them strict rules by any means.

With regards to rudeness of usage of get + preposition, I would say again this is really context specific and largely depends on the tone of the speaker's voice. If i had to make a rule out of it, the negative commands "get out/get down" are almost always rude. "get in/get on" doesn't have to be, unless the speaker sounds angry.

"get down the chair" is also not correct. "get down from the chair" is better but best would be simply "get off the chair." But again, this is something you would use with a small child or a dog. With an adult this would be rude, instead you'd ask them "can you please stand up?"

Answer

You've identified more or less the difference between "come" and "go".  "come": from there to here. "go" from here to there, or from there to somewhere else.  Prepositional verbs which use the literal meanings of "come" and "go" follow this.

Small mistake: "get off or get down from the chair."

Climb on(to)/off a chair is also good, but a little more formal than get on(to)/off a chair.

Enter = go/come in   Leave = go/come out of.  They are more formal.

None of these verbs based on "get" are rude, just informal. If you use them to order someone to do something e.g. get off the chair! , then the other person might think you are rude, in certain contexts. But all language can be used to be rude, if you want to be rude.

Answer

You're exactly right about come and go.
With 'get' it doesn't matter where you are, and the direction is usually obvious from the location  -  you get in/out of something, usually something smaller. 
Get can be rude, but only if it is used as an order, or used in a rude way. As long as it is not clearly an order, or said rudely, it's fine to use.
"get on the chair" = "climb up ON the chair"? <- not really. There is no 'up' implied. It could also be a low chair, in which case 'up' is not valid.
"get down FROM the chair" = "climb down FROM the chair"?