I would hang on to them for a year, just in case any any questions arise, like whether you deposited a certain check or not. Usually the record that you paid someone else through a checking is the most important record.
the general rule of thumb is after 3-5 years it's okay to dispose of them. If you're paranoid though you can save them for longer.
it's suggested that you keep every financial document for 7 years.
yes
Put them through the paper shredder.