First off, thank you for your service. I'm sorry you are in this situation after believing that you paid off the loan. From the info that you have provided, it sounds like you defaulted on the loan in 2007 - then in 2008, you made a large payment that you believed would settle the loan. Then nothing happened until your 2012 tax return was garnished, correct? And nothing since - no wage garnishment or collection calls? I assume that this was a federal student loan.
First off, have you called Sallie Mae? If they no longer own your loan, you can look it up in the national student loan database (the name may be slightly different, but it's a Dept of Ed website). The database should include all of your federal student loans, their current status, and the current lender that services the loan). If the status is default, your tax returns will be garnished until the debt is paid). I would contact the current lender or the Dept of Ed directly and ask about rehabilitating the loan. Perhaps someone else will chime in with their experience - I'm actually not sure whether rehabbing a loan will stop garnishment, but it will definitely help your credit, so that's what you want to do if you can!
As for the question of whether the defaulted loan will be a problem when applying for a mortgage, it is a major derogatory, so my guess is yes. I would try to rehab it - though you may have to push back the home-buying plans by a few months, since rehab takes nine months, plus whatever time it takes to update your CR.