1

You don't mention your citizenship so I'll assume US. A tourist visa is good for 180 days. Getting a 30 day extention or a visa for 1 year takes time and proof of income.
You will need a return ticket in order to get a tourist visa. Your airline will insist on this.

good luck!

2

Depending on the climate you wish for, there are many options.

Concepción or Bugaba in Chiriquí would be good. That would allow you access to the Western Highlands as well as Costa Rica, Puerto Armuelles, and Bocas Del Toro. It is quite a ways from Panama City.
If you wanted to be in Chiriquí but in a cooler climate, you could try Boquerón or Dolega.

All of these places you could rent a place for much less than your $700.00 budget.

Also as alternative locations, there are a number of places in the Azuero Penninsula that could be interesting to stay. Las Minas is a bit on the cooler side, or perhaps if you want by the water, Tonosi, or Pedasi.

Pesé has a vary alternative lifestyle attitude. Very gay friendly... also real cheap compared to many locations.

There are also some areas near Panama City that could be fun. Even Chorerra could be ok. It is close to the city, but is in essence a large city itself, albeit working class, but along with the working class people, comes less expensive living.

Or you could even try places a little wilder, like Veracruz, or Portobelo.

Regardless of where you ultimately choose, it is alway smart to check out your neighbors. Panama is a very social country, and good or great neighbor will make your living experience much better.

Oh, if you wanted to try some island life, you might even try renting a small place on Taboga Island.