The coup in Egypt happened because the muslim brotherhood were threating the power and wealth of the military. It was nothing to do with protecting democracy.
No they don't military coups happen when corrupt military leaders think they can grab control of a countries assets or aren't being paid off enough by weak dictators.
No, it wouldn't, but your comment does illustrate the problem with this discussion, i.e. that people tend to argue for simplistic magic bullet solutions ( apologies for the horrible pun). To deal with this the US needs to look at a variety of issues, from gun control to mental health provision...