I have special interest in know how they come up with the best joints for the best task, like "oh, i need x degrees of freedom here, but it needs to support y weight, so since Im limited to materials s and z, I think joint j will do the work".
Is it mechanical engineering? is it industrial design? is it not really present in a single field? I mean, robotics is pretty generic and deals with a million of different stuff.
So, say a team is making a robot, say something like big dog, how/when exactly the joints gets decided..?
Those questions usually fall in the mechanical engineering (ME) discipline. ME basically touches all other engineering fields and some of everything's in there. You're right that robotics is something that crosses a lot of disciplines, but your questions about joints and materials would be ME all the way.
You've actually described at a very high level what engineers would do. First, you have to define your functional requirements. This is usually like a use case, or some specific set of "must-haves" in the design. Sometimes that may be the total degrees of freedom, but sometimes that's simply "the machine must be able to do X or Y". This usually frees up the constraints in the concept generation phase. At that point, you come up with a basic design that may work functionally. You check that design over to see if there are any obvious problems with it, then you start analyzing it. There are lots of different analyses you can and should do, such as kinematic (to see range of motion and maybe if it can crash into itself), structural (to see if it can hold itself up or if the material will yield during operation), joints (like what kind of torque you need to move the robot armatures around), tribological (to see if friction/wear is a problem and where it is), tolerance (to see if the tolerances on different subcomponents are too tight or too loose for assembly), failure safety, motion repeatability, and so on. This is the basic design process. If at any point you come across a problem that is "unsolvable" (i.e. the joint type causes stress peaks that yield your material or the parts cost too much), you've got to go back and redesign. The further down the process you go, the harder it is to go back and redesign, so usually following rules of thumb help you save yourself early on. You don't always have to do those kinds of analyses. If you're trying to build a one-off or a prototype, you may not care if your tolerances are defined for general manufacturing. It does help, but not completely necessary. It just depends on how you're comfortable proceeding.
Mechatronics is kind of like a hybrid of mechanical and electrical engineering and deals with the application of electromechanical components. I would think a lot of the knowledge of servos/motors/etc. would be there as well as in electrical engineering. AFAIK and I was told, electrical engineers don't deal too much with this type of thing anymore because they do a lot more signal processing and systems design (the kind with complicated ODEs and filters and things I don't fully understand).
If you're interested specifically in how joints are studied and forces calculated (roughly), then I would suggest "Design of Machinery" by Norton (ISBN: 978-0077421717). " It's not the best book on the subject, but I know what's in it since I used it in my kinematics class way back when. Once the kinematics is explained, it goes into how to calculate forces. Check it out or borrow the book (or PDF?) from somewhere.