Are you in the US? It should be 8 years/100K miles. Unfortunately, last time I checked, the Prius c doesn't get a longer 10 year/150K mile HV battery warranty in CA and CARB states. You should be able to download the warranty booklet if you select the Prius c in the upper right of Toyota Online Owners Manuals and Service Manuals.
How often do these batteries need replacing, if they do fail, and what kind of driving conditions are more taxing on the battery?
Don't know. They can last a long time. See Lifespan/Operating costs - Prius Wiki. Toyota has frequently said life of the car. They've also had numerous brochures (like http://www.toyota.com/byt4/2012/priusc/en/complete_ebro.pdf that I just found) that says "Hybrid vehicle battery expected life is 150,000 miles based on laboratory bench testing." Google for site:toyota.com "prius c" lab testing battery. I wish I could find one of seilerts' excellent posts on the subject and our hypothesis, but 2005 Prius 241,000 Miles Hybrid Battery Replacement Story | Page 2 | PriusChat might be good enough. Personally, if possible, I'd park in the shade, cooler areas (one that gets less sunlight in the day) in the summer or covered parking. Your Prius c's HV battery is warranted for 8 years/100K miles.
my dealership says they have never had to replace a prius battery out of warranty, and only 2 in warranty due to defects.. so.. and they she said, by the time my warranty is up the battery in my car will cost prob about $400 at most the way the costs are dropping for that technology
Do you believe everything your salescritter says? You should not. Battery prices have dropped, but are not falling anywhere near that dramatically. And PC has had numerous reports of battery failures out of warranty, though not really at any rate worse than non-hybrids need transmission work. Many of them have been directed to used and refurbished battery suppliers instead of new units at dealerships. My Prius replaced a car I drove 235k miles. I am budgeting to replace the traction battery once in that distance. Numerous owners who drive many miles per year in not-too-hot climates have gone well past that on the original battery, but some others with fewer annual miles and hot climates have had failures beginning shortly after the warranty expired. No one has provided us a statistical analysis.
So what'll happens when the batteries die? The car won't run or will run on gas engine only? I guess the computer won't let it run on gas only, it would be interesting if when we don't have money to replace the batteries we just use gas engine only.
Yep. OTOH, the symptoms of a bad HV battery (at least on Gen 2 and Gen 1 Priuses) include the battery fan running on high, poor mileage, the SoC jumping around (goes very quickly from empty <-> full), warning lights and IIRC, poor acceleration and engine revving high as well. The HV battery is what starts the gas engine, not the 12 volt. If the HV battery goes bad, it'll need to be repaired or replaced. See Failed traction (HV) battery, what to do? - Prius Wiki. I doubt a NEW replacement HV battery for a Prius c will drop to $400 by the time it goes bad.
My 2007 Prius hybrid battery died after 6 years (and change) and 95k miles in SoCal. Being replaced under warranty now.