I believe Begründung = the act of providing reasons for, to found, or justify whereas Gründe = reasons (which are not necessarily the reasons we invoke). You might have done something for a certain reason (Grund) without specifying that as your reason for doing it (Begründung).
Not really, although the meanings may overlap in some contexts.
The reason the bridge was destroyed is that a new one needed to be constructed in the same location. The cause of the bridge's destruction was the explosives set by the engineers. The issues surrounding replacing the bridge were discussed for over a year by the local community.