U.S state government initiated H1B visa

Hello Friends,

Please respond my question if you know the answer.

I am in F1-OPT EAD and eligible to work till Dec/2019. I have been selected to a IT position

as permanent employee with low pay (< 40K/Year) in US state government agency/department.

My employer is ready to file H1 and once approved to file GC as well but I have to pay all petition fees.

With my employer I am the only one foreign worker, my employer doesn’t know much on this me only

have to educate. Now, below are my questions in this case.

Is the H1-B will go cap/lottery or cap exempt or is any special quota/process for such petitions .?

Considering me only the foreign worker with my employer, how much is the H1-B petition fee .?

Is the GC process also same like general way or any special case to get approved in short time .?

How much expensive is the GC petition .?