Done smartly, living in an RV can be relatively cheap, though a cheap apartment can compete in many areas.
Most long term sites have you paying for electricity and you have to pay for your propane.
- 6 months summer in Michigan our bill totaled $176. Texas summer would be higher but probably not more than double.
- In winter, you will probably go thru a 30lb tank of propane every couple weeks unless you hit a cold snap in which case you will burn propane faster. In -5F, we've gone thru a tank a day but in 15F, it might be a week. Hard to say exactly as how much you open the doors, thermostat setting and wind can significantly affect it.
If you are on a budget, don't worry too much about 4 season. It's nice but not critical. Unless you go big (probably not an option on a budget) a standard air/con should handle the heat OK.
10yr rule has nothing to do with durability of trailers. It's just a way they can screen out old beat up units. If the trailer looks nice, they forget to ask the age. If it looks bad, they ask the age and then tell you it's a rule.
$5k is realistic but you will have to shop around and really check out things before buying. That should get you something in the 25-30' range about 10yrs old. Biggest thing to look for is water damage. After that make the plug it in and run every appliance and plug into every outlet. If that all turns out good, you are probably fine.