GATE

Eligibity criteria for appearing on GATE is that candidate must have completed Bachelors degree in Engineering/ Technology (4 years after 10+2 or 3 years after B.Sc./Diploma in Engineering/ Technology) and those who are in the final year of such programs. Even if you had backs, you are eligible to appear for GATE.

Based on your profile you are eligible to seek admission in some German universities.

They mostly rely on GRE, based on GRE score, it's good to join MS industrial engineering. They prefer GRE and their test for MS in Germany.

Firstly, clear all your concepts from the subjects in B.Tech EEE and stay focused. Go for Made easy (GATE coaching institute) take the test series or classes on weekend or online lectures that would be suitable for you. The questions in GATE are based on the logic which can be cracked by through in-depth study.
Good luck.

Eligibility criteria to appear in GATE is Bachelors degree in Engineering/Technology (4 years after 10+2 or 3 years after B.Sc./Diploma in Engineering/ Technology) and those who are in the final year of such programs. During your B.Tech Manufacturing Engineering you will be covered majority of topic included in Syllabus of Production and Industrial Engineering. You are advised to compare your college syllabus of Manufacturing Engineering with GATE Production and Industrial Engineering syllabus and do self preparations in gap areas if there are any gaps.

Yes, you can attempt for GATE now and join the course in the next academic year.
Or
You can attempt GATE after your current B. Tech course completion means next academic year.
You can attempt GATE maximum Three times.
The GATE score valid for three years from your appearing year.

There are so many coaching that it cannot be easily said which are best until we have a clear cut knowledge of the study materials, teaching skills, faculties, etc. "Made Easy" is much well-known coaching class with its centres in many cities.

As all subjects don't have equal marks, few subjects are preferred over others
Make a list of topics which need to be studied based on the number of questions which have asked from the topic in the previous year's GATE examination.
Identify topics which can fetch more marks and give more importance to them during your preparation.