It depends. Do you mean like learning at the same time while teaching the student? (answer would be no) Or do you mean after you are done teaching and got some free time to learn another language if so yes.

Not only you can, but I also believe that those language teachers who are students themselves have a great advantage. This is because they understand the difficulties and all the challenges that their students face (motivation, consistency, etc.). On the other hand, being a language teacher will make you a better student because you can see the things from the teacher's perspective as well. So, my suggestion is go ahead!

Sure you can, for example I am learning German and English and I am teaching Italian :)