I definitely think you should get both of your reports. Don't worry about them fishing for any information; they are only trying to verify information that would already be on your reports. Just make sure you are at the official websites for Transunion and Equifax and not one that is fake. You need to know what is on those reports and if anything is inaccurate or fixable. Late payments can sometimes be removed by way of writing a "goodwill" letter to the creditor, see this thread for some examples of letters I've written in the past with success:
/threadview/2186.html
If I were you I would leave the HBC and Home Depot cards open. Closing them will reduce your available credit, which in turn will negatively impact your credit score. (At least in the short term.) Probably not by very much but it's just easier to leave them open and never use them, or even carry a small balance on them (ie: less than 30%).
Again, you won't drag up anything old that isn't already there. In fact, there may be items on your report that ought to have been removed by now, TransUnion is notorious for "forgetting" to purge derogatory items that have been around for longer than 7 years (or whatever the time limits are in your area) and sometimes need a reminder.
Good luck!