I don't know when they became sacred (and according to Cicero in Cic. de Div. ii.34, any bird could be used, but chickens were preferred) or exactly when the romans stopped believing in such matters, but it would have most certainly been during the decline of Greco-Roman polytheism where religions such as christianity were being brought to the fore which required no sacrifice and thus no sacred animals.

Take a look Here for some info on it, perhaps searching for "Sacrifice" within the page. (And yes, I know it's not the favourite source for the sub, but it should provide some idea, and lead onto further reading if necessary.)

As for your main question, no, the Romans still ate chickens, however some laws were put in place to stop this, such as in 161 BC, where a law was passed in Rome that forbade the consumption of fattened chickens. It was renewed a number of times, but does not seem to have been successful - evident in a Roman cookbook by the name of Apicius, which offers 17 recipes for chicken.

Chickens were used in a kind of divination. The chickens were released from their cage and thrown some food; if they ate calmly it was a good sign, if they weren't hungry or made a commotion, it was bad. This form of divination (the Romans had many) was usually used before a battle or military endeavor to determine whether conditions were favorable.

In earlier days they might use any kind of bird, but eventually they settled on chickens because they were easier to raise and handle.