Hello Dear

Yeah you can able to give from these branches but possibly try to give through your your branch only such that it will be easily for you to get into the PSU under your domain as well from that branch which you have done your UG Degree if you apply the GATE from that branch then easily you can able to do the M.tech from IIT / NIT rather from other branches.

All the best

Yes you can definitely appear for GATe ECe paper for ME/MTECH courses.
Entire eligibility criteria shared below.

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

https://engineering.careers360.com/articles/gate-frequently-asked-questions

Graduate Aptitude Test in Engineering

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