The answer to the question how many miles will x car last is... It depends. Clearly some models have better reliability that others and every brand has good and bad models, but how it was used and maintianed by previous owners
Here are my tips
1. Use the reliability data available to you, to identify which model to consider. Don't just use the brand perceptions as many do to make decisions. I use this site andhttp://www.dashboard-light.com/
2. For the best value Hit them where they ain't - look for models that are not popular - i.e. smaller cars vs crossovers, Minivans vs 3 row SUV's etc, or models from dead brands (that still have protection i.e. Mercury, Saturn etc)
3. Think about your future needs - If you plan to keep a car for say 5 years - do you expect your requirements to change - Will you family grow? Will you likely change jobs and have a longer commute?
4. When you find a car learn as much about it's history and condition as possible. Carfax is a start, but and absolute must is an independant inspection - or at least a friend who knows what ot look for. I prefer to by from private indiviuals as they will know the history. Most dealers / auctions distroy any history records under "privay concerns". Find out where the vechile was previously owned and if possible try to figure out where it was serviced - contract the dealer / garage to see it they can provide you the records
In terms of your spec above consider...
Toytoa Prius - I brought a 2010 (gen 3) last year from my neighbor for just over $6K. THe milage was ~ 140K but drives like new - these are renound for going 250K miles without too much issue. It's a boring but easy drive. They are supprising spacious inside and easily get 45 - 50 mpg. Gas price is low ans so Hybrids are less popular right now.
Kia Fotre . A 2011 - 2012 is right around your price point. You could also look for a kia soul - Look for the 2nd generation
For an off the wall idea What about a Smart Fortwo - you can get a 2014 / 2015 with balance of the factory warranty for well under $6K
Another area is larger cars like Chevy Malibu or 2nd gen Mazda 6 a 2010 model is in your price range.
I hope this helps