People have had varying degrees of success with calling the servicers for the accounts that are still showing late payments from before consolidation.
Personally, I've had 0 success with it, but it doesn't hurt to try. If you search this subforum for "contact info" or "goodwill letter" or similar terms, you might find some other threads that have specific emails and phone numbers you can try.
Lates from 5 years ago are going to hurt you with home loan underwriting a lot less than you might think, so good luck.
Getting all revolving credit down to 0 balances on your credit report will boost your score a bit and also most likely increase the amount you can get approved for with a home loan. A big factor with automated and live underwriters is how much you pay per month TOTAL on any open loans or revolving credit you have.
Good luck! Check out the home loan subforum!