An Indian citizen having a Green Card can he work as a Govt. Employee in India even having a Green Card.

I don't see why not, unless it's a super sensitive military operation. I imagine that India is similar to the US in that regard - each division and agency has different security clearances.

Note that working abroad in any job for a long time could indicate you abandoned US permanent residence by failing to maintain residence in the US.

Yes that's possible

You cannot live outside US over 12 months, your green card will be automatically revoked.