@Justin D. --
There are lots of loans with good terms for first-time home buyers, loans that someone who already has a mortgage won't have access to, like the USDA and VA loans, which are 0% down, FHA loans which are 3.5% down -- and sometimes you can get a HUD home with a FHA loan for $100 down (that's what we did). Quicken has a 1% down program, and lots of other banks have loans with small down-payment requirements. Some loans may include repair costs in escrow, so if you find a house that needs work and you can get it at a price well below what it's after repair market value would be minus the cost of the repairs, that could be an option. Just keep in mind the lender is unlikely to release the funds to you all at once before you make the repairs, it's more likely you'll get reimbursed for repairs in stages (called "draws") after you complete them. The loan might also require the work be done by a licensed contractor, so read the fine print.
Some states and cities also have first-time home buyer down-payment assistance and forgivable rehab loans for certain areas and certain incomes.
Different lenders have different requirements regarding property condition. You can search on Google with terms like "USDA loan property requirements." A knowledgeable mortgage broker will be able to tell you which loan would work best for the condition of the property you're looking at.
Two loans good for rehabbing are the FHA 203(k) and Fannie Mae Homestyle. These loans let you borrow for more extensive repairs. Ask your mortgage broker about them. Some of the work you may be able to do yourself, but again, read the rules.
As for mortgage insurance, the fastest way to get rid of it is not to get it in the first place -- put at least 20% down and you won't need it. But that's not what I'd recommend.
In my opinion, the best way to get rid of mortgage insurance is to get a great deal on a house, then refinance once you've got 20% equity. It's straightforward. All you need to do is find a bargain when you buy the house in the first place.
With a really good deal, when the total costs of your purchase and any repairs you'll need to make won't exceed 80% of market value after repairs, you've got 20% equity as soon as you fix it up. Now all you have to do is wait 6 months, and then you can refinance. (Some lenders might let you drop the insurance and keep your current loan after a new appraisal, which would be a much cheaper way to do it than refinancing -- something else to ask your mortgage broker about when you're loan shopping.) Boom, no more mortgage insurance.
Really you need to shoot for total purchase and repair costs at less than 80% market value to account for the costs of an appraisal and closing costs for the new loan. Due to these costs, it really only makes sense to refinance out of the mortgage insurance if you're going to hang onto the house for a few years. Otherwise, another great way to get rid of mortgage insurance is to sell the house and take whatever equity from it into your next house.
Of course, there are interest rate considerations, too. By the time you're ready to refinance, the interest rate for your new loan might be higher, and it might make more sense just to keep your mortgage insurance until you've paid down enough of the loan that it drops off.
With your credit score, you might qualify for a loan right now, depending on your debt-to-income level. You don't have to have zero debt or a perfect credit score to get a mortgage -- I'd be surprised if there was a significant difference in interest rates between 780 and 800.
My unsolicited advice would be, and I'm sure there are many who would disagree -- hang onto as much of your cash as possible when buying a first home. Get into a house as soon as makes sense for you, as cheaply as possible in a good neighborhood that might experience some appreciation, and if you're single or your family is up for the adventure, go for a duplex, triplex or 4plex instead of a single family home. Take the money you didn't end up using for a 20% down-payment, any cash flow from the other unit(s) in your house, and maybe even the cash you didn't use yet to become debt free, and -- assuming you already have a 6-month emergency fund set aside -- put your surplus into buying another income-producing property. At this point you'll likely need to put at least 20% down, 25% for a much better rate. Not doing that on your first home could put you miles ahead in building wealth, because you can put that money into an additional passive income stream.