I added Freon to my 2007 Chrysler Town and Country van, but the air stayed cold only 1 day.....now what?
What can I expect the costs to be for repairs, replacement of parts, etc.?
As a woman, can I get free written estimates?
How can I prevent someone from misleading me?
You obviously have a leak in your ac system. As far as a free estimate, you may get a free estimate for the cost to find a leak. (Proper diagnosis). If I was to guess, the first place I'd look is underneath at the rear of the vehicle where the AC lines go into the rear evaporator. This could get costly, and know that there are kits to just eliminate the rear without replacing the lines and/or the rear evaporator if it is indeed the source of your leak. The downside of this is your passengers in the rear may feel like 2nd class citizens, sweating profusely in the back, while those up front need to turn the AC down because they are getting too chilled.
It really depends on how willing you are to get your hands dirty. Based on your description its almost certainly a leak but the basic system appears to be functioning, hence why you got cold air for 1 day. A lot of the consumer-grade refrigerants have UV dye in them -- It will highlight the location of a leak. Sometimes you can see it with your naked eye but they have UV flashlights that really make the dye pop out, these lights are readily available and not expensive at any auto parts store. Specific to your vehicle I don't know what the common failure points are but follow the AC lines from the compressor to each component as best you can, with any luck the leak will be obvious if it only took one day for it to leak out.
If the refrigerant you already added didn't have UV dye, then buy a can that does and let it leak out, and look for it.
I assume the air blows warm all of the time, if it blows warm only when the vehicle is stationary (but blows cold while moving) then it's your cooling fan and has nothing to do with your air conditioner.
If you're not comfortable doing this you're unfortunately at a mechanic's mercy. However I would be wary if they recommend something like a new compressor (which is probably not needed unless that's the source of the leak). Ask them to show you the leak. Also note that R-134a refrigerant is very inexpensive to a commercial garage - You can get 30 lb containers on Amazon for $162 shipped (or about $5.40 per lb, your vehicle probably takes 2-3 lbs) and it will be even cheaper for a commercial business. I find that less than honest establishments will grossly overcharge for refrigerant. (For my home air conditioner I was charged $300 for 2 lbs of R-410a refrigerant - almost 30x their actual cost. Needless to say, I used a different vendor next time.)
Personally I have had good luck with AAA Auto centers and they seem to be consistently decent if you happen to have one nearby.