Based on the criteria you are looking for there are going to be options available to you out there.... I have sold a few homes to people doing the same thing you are looking to do. They bought a house cash and did the minimum necessary to live in the house and worked on the rest of the house as they lived there to ensure they did not have a mortgage. At the $15k mark you might find a home in a lot of disrepair but in the $30k to $40k range you should be able to find something that needs a little TLC and help along the way to get it to what you want it to be.
I would be happy to help you find the right house for you and your family. Feel free to e-mail me or call me on my cell.
Helen Adams Realty
If you are looking on line and think you have found something good…. do a drive by. Many are in neighborhoods you won't want to buy in. Most foreclosed banks do their research. So either it's been on the market before and didn't sell at "market value" or it is now priced right… at a cheap price and will get multiple offers from investors. Make sure you know the areas real value and make sure their isn't something wrong with the house you can't fix. (i.e. power lines, bad neighbors, steep hill, sink hole, septic issues).
Make sure you have gotten good estimates on materials and the cost of jobs you won't be able to do yourself. Also check out the local building codes. Many may add additional cost to your plans.