As per the GATE eligibility, Currently in the final year student can apply for GATE Entrance Exam. You can check the further eligibility with the given below link.

https://engineering.careers360.com/articles/gate-eligibility-criteria

Good Luck!

Yes you can if the PSU allows it and many do. so please look for those PSUs and mind you they give time till August something for you to submit the marksheets etc. So go ahead and prepare well. All the best!