While driving the car, when I apply the brakes or give left/ right turn indicator, then the 'green drive' light in the dashboard goes off for couple seconds and then comes back again. Sometimes, the check engine light also blinks for a second along with green drive light. This is for 2004 Honda accord, EX, 2 door coupe.

Since this is out of warranty it is going to be on your dime. I would not jump on this until you blow a fuse, I think you have a sporatic short

Pretty weird. I really don't know what to make of it. Have you checked for any codes?

The mechanic has replaced the battery with a new one. It ran well for 50 miles then the problem started agaim. There are no error codes but he said that few monitor are off

What does "few monitor are off" mean

There were no error codes displayed during the diagnoistic but he said that three monitors are off / not ready and it will not pass state inspection. He asked me to drive another 50 miles when the problem started again. Also, if the speed of the car is too slow, the car will just stop, I will have to start the car again and then drive.

Diagnostic might need to be done while driving the car,perhaps it's on a moving part that has the problem which isn't readily apparent while in park.

The gas warning light comes on before it hits 'E'. I see that there is 2 to 3 gallons odf driving left before the fuel indicator will hit 'E'. Is this a problem? Are both the problems - gas warning light and green drive light - related?

Probably not, the warning light for the gas is supposed to come on when you are running low,no running out completely(there would be little gained if it tells you your out when your out,right?)

Have you gone to more than just one mechanic to remedy this?Opinions like certain body parts,everyone has at least one(perhaps 2)

My initial reaction would be to inspect the brake/tail lamps for filiments that have cross-shorted, creating voltage feedback into other circuits.
You could also have a poor ground point somewhere along on of those related circuits.
Hands-on is the only recourse with this car.