No reinstalling office will not bring "Works" back, they are seperate products. Try looking in the c:\program files\ folder for "microsoft works." If its still there the shortcuts might have been deleted and you can just re-create them.
My 2nd question is what do you need MS Works for, that office 2010 will not do?
The program in your recycle bin is Microsoft Security Essentials (64bit) thats the antivirus application you installed.