GATE

yes.there are colleges.for details contact czero7777@gmail.com

no you required BTech degree

after HS?
In first year you should cover the engineering mathetics in your course work according GATE syllabus. After first year you should solve the last ten year papers from covered portion first according to your course work.

Mechanical engineering in public sector however no job is offered in private sector on the basis of gate score but definitely you will get some edge in your selection

Go for M.Tech as it will help you to choose your career in academia in future

You should focus on your first year math only in B.Tech first year for GATE later on self study of course and solve the numerical as much as you can. Also solve the last ten year papers of GATE.

Distance education is not allowed in engineering but you have to attain the practical. For example B.E is the part time course whose classes are scheduled in the evening and practical work on the weekends. For more information you can visit any IIT website for eligibility criteria. You can found it on GATE section. Many information is already available in previous answer.All the best.

I think it is not a good option however you can join test series of it. Test series will help you a lot. You can download lot of videos from you tube and NPTEL of your engineering subjects.