Do cooked rice and/or grains get bad fast without a refrigerator?

No more than an hour really. Cooked rice and grains can "sour" and spoil very quickly if left at room temperature. They're pretty notorious for growing bacteria.

Uncooked, however, grains and rice can last years at room temperature if stored correctly.

yes because they are very low acid.
I would not leave them out more than a couple of hours.

no more than a few hours - especially for rice, there's a mycotoxin bacteria that isn't killed during cooking

They will turn sour if the temperature is warm enough... Its alright to leave at room temperature but remember to allow it to "breathe". If it is not properly ventilated. It might turn sour too....

yea they smell baaaaaaaaaaaaad ._.