No, it shouldn't leave you stranded.
And yes, the symptoms you describe regarding the t-stat sound exactly like it is stuck open. The temp needle should never EVER go down from 12 o'clock. If it does, the car is over-cooling itself due to a t-stat which is stuck open. Same exact thing happened to my car. The t-stat stuck open, which caused the temp needle to go down, with less heat coming from the vents, and it eventually tripped my check engine light due to the engine running rich (02 sensor code). That may likely be what is causing your catalytic converter codes.
As far as how much it will cost you to replace, the part is about $25, but you'd be well advised to also replace the t-stat housing at the same time, which I believe is about another $30-40. I have no idea what the "book rate" is for this job, but it should take a skilled mechanic no longer than an hour to replace, and many mechanics charge between $80-120/hour. You'll also have to pay for some coolant to fill the system back up after the new t-stat has been installed.
Last edited by ZeGerman; 01-25-2013 at 04:51 PM.