Any dealership should give you at least a 1000 mile/ 30 day warranty. That should give you enough time to see if it has any problems. If they don't want to give you at least a month's warranty, then they're selling you something they know is bad.
If the dealership told you they bought the car at auction, then I don't see how they could've gotten a history on it other than from carfax. I see you did a carfax on it already, so if the dealership told you something more than what's on the carfax report, then they are either lying or who knows what.
Personally, I would rather have a CE brand new off the lot than an LE with close to 40K miles on it. For just a bit more than what you're willing to pay for this car, you can pick up a CE brand new at a toyota dealership. I paid $13,200 after taxes for mine.

It was $11,999 before taxes.
Power options, alloy wheels, and all the other things that an LE would typically come with aren't really things that I care for. In fact, I prefer not to have power windows/locks because they make after market stereo installation that much more of a pain in the ass. Also, I like knowing that I can manually open that shit up if I ever need to. I don't feel comfortable with the idea of the locks and windows being controlled by an electrical power source.
Although it's still an option for the LE, anti-lock brakes are really the only reason I'd consider an LE over a CE.
Since you are new to California and probably to the United States, you probably are not aware of lost leader ads.
If you flip open the newspaper on a friday, saturday, or sunday, you'll see big, huge, colorful dealership ads with rock bottom prices on new vehicles. They'll say "1 at this price" or "2 at this price."
The problem with those deals is that you have to get to the dealership on Friday morning right when they open. If you get there later in the day or the next day, they will tell you they've all been sold.
I've been there done that twice with lost leader ads. That is, I've successfully purchased vehicles listed in those ads on two separate occasions, so they're definitely real.
Let me know if you're interested in swinging that. I'll give you a more detailed run-down on how not to get raped by the dealership.