Is it possible for me to work at NASA or other research companies as aircraft mechanic?
Probably not, unless you are a U.S. citizen (dual-citizenship should be ok.) According to NASA,


  Other than under extremely rare exceptions, you must be a U.S. citizen in order to work for NASA as a civil service employee.


It's extremely unusual for non-citizens to be able to work in the U.S. in agencies like NASA or the Department of Defense, where there is necessarily lots of equipment and/or information subject to ITAR controls and/or which is classified. This will generally also be true for contractor positions. 

At my present job, we are not allowed to disclose anything more than high-level marketing material regarding products subject to ITAR to anyone who is not a U.S. citizen, for example, even though we don't make weapons or anything like weapons.