Can I get a government job after opting for B.Tech in CSE?

Majority of engineering jobs in government are for civil, electrical, mechanical and ECE engineers. There is no direct cadre for CSE engineers in central and state governments. You can join a government organization such as DRDO, CDAC, NIC, defence services, paramilitary forces, ISRO, civil services

Yes, there are many intelligence bureaus which are hiring CSE engineers for programming. If you are good in hacking and coding then you can have a great opportunities in government sector.

You can but most of the time GATE qualified students are preferred. CSE is mostly for entry into the corporate sector.