Once again, there are several ways of saying the same thing!! However, Pelin, as Business English Ed points out, you have to add on after up (in your first example). This is because you can't get up one side of the bed. At least, we don't say that - though we do say you can get up one side of a mountain. Anyway, here the idea is that you get up, and, secondly, it happens on one side of the bed. So the two prepositions although not really together appear one after the other in the phrase ...... you get up (you don't get up one side of the bed by climbing - you just get up- where does that happen....on one side of the bed. Getting up one side of the bed would suggest climbing up the bed. But here, no. You are getting up (yawn etc.), but damn it you got up on the other side of the bed, not your usual side.
With "What side of the bed did you wake up today" that one is wrong too, because you never wake up the bed (you really wake up yourself). So if you don't wake up the bed you don't wake up one side of the bed. Make sense?
And the idea from the expression is that because you got up on (and then got out of) the wrong side of the bed this has for some unfathomable reason put you in a bad mood. You can't make this shit up! It's such a stupid expression.
I remember seeing this cartoon of a guy wearing high-heeled slippers with feathers over the toes, and 2 guys standing beside him say, one to the other, "What's happened to Joe?"
And the other guy answers, "Oh, he just got up on the wrong side of the bed." Hence he put on his wife's slippers. It's funnier when you see the cartoon!