Hello!
First time using sketchUP, so im a complete newbie.
Somehow i am unable to move things vertically in a correct way.
Clicking the up arrow did enable the movement, but it moves other lines as well, which i dont want…
Anssi is correct that you should familiarize yourself a bit with the basics.
Based on the screenshot you are sharing, I am guessing that most of the modeling you have done is what we would call “raw geometry” and that is most likely the cause of the issues you are seeing when objects are sticking together. Remember, groups and components are your friends when it comes to complex models.
Additionally, selecting multiple objects can be done a number of different ways. If you show us a screenshot of what you are trying to select I can offer some advice on that too.
Finally, there is a way to separate the rectangle from the rest of the model, but it again involves groups and components so make sure to read up on that.
There is a way! Select the group and its edges by double-clicking on it, then create a group to contain that face. That will isolate the geometry in the group from interaction with geometry outside the group. As @Anssi wrote, this is basic SketchUp which taking some training should clear up for you.