It depends on your state for example in Rajasthan the Rajasthan technical university is governing body of all state level engineering colleges and it don't organise a separate entrance exam for entrance in those colleges. It considers our jee mains marks for college allotment while in west bengal wbjee is the criterion for selection in their state colleges. So i will suggest you to check what your state colleges considers for entrance into them.

Best if luck!

Yes, there are many state colleges that intake students on the basis of their jee ranks

Yes you can get admission in state wise engineering colleges.