1- Having more lines of credit (in good standing) is a good thing, so leave them if you want. I personally closed out my inactive credit accounts not because of my credit score, but because I never used them, and it's just another potential target for fraud/identity theft.

2- You can't always make old marks go away, but sometimes you can. Call up the creditor with whom you made a late payment and politely harrass the shit out of them. Tell them you have no record of the late payment, and you would like to see their records and verify it. Sometimes rather than dealing with an annoying customer, they will remove the record. Kind of luck of the draw whether this will work.

3- ?

4- No. Applying for credit cards actually hurts your record in the short term. Applying for cards you never intend to use is also not smart. I'd start (responsibly) using some of your older accounts rather than starting up a new one.