GATE

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.

Hi,
You need to have 4 years Bachelors degree or more to be eligible for GATE. Check here all the other options:
http://www.gate.iisc.ernet.in/?page_id=130

you can join any online institutes for preparations of GATE & there are more books available just study hard & you can give GATE exam only 3 times in your career.
go through some classes.I'm also pursuing BE in IT engineering & I'm preparing for GATE from GATE India forum the tuition classes. They have provided me the material & lectures regarding GATE which make me East to prepare. & If have query just Google it. Google is the best option for any query or doubt there are some videos regarding GATE preparations on you tube also.

Hi
For Master's at IIT, you need to appear for IIT JAM and GATE is also accepted. For abroad, you must have GRE/GMAT score and IELTS/TOEFL. Rest of the requirements will be as per the country you choose to apply. You may go for M.tech/ M.E / Mba abroad. For USA, take a look here.
http://www.masterstudies.com/Masters-Degree/Engineering-and-Technology/USA/