Graduate Internship - Collaborating with other divisions
1- do you think that it would be appreciated? Or will I sound as an annoying climber?


No, I personally would not think that. Instead, I would appreciate your initiative and passion.


  2- do you think it could upset my first division? (provided that I'll give them full priority, doing parallel activities ONLY in my free time)


Here it comes the important point. It is ok to collaborate with other groups but you should ask your supervisor first. Even if we are talking about your free time it could still sound very shady to work with other people without saying anything. When you ask this to your primary division, you can say what you said to us "I would like to collaborate with lab X in this way to improve my CV, for networking [...] HOWEVER, I will give full priority to your tasks [...]"