Is it true that rice fields are a big producer of methane gas and do more damage to the enviroment than cars?

Answer 1

no.
the biggest producers of methane gas are cows and sheep.
the next biggest producer of environmental damage is the transport industry.

Answer 2

No . Not at all. Rice fields do not produce any methane. In some countries cow dung is used as manure. Cow dung after decomposition yields methane. But this is well harnessed as cooking fuel even in small villages.