Punjab

Yes, it is necessary for you to give an entrance either its JAM or JEST because IISER offers integrated Ph.D. programme. Here is the link:
http://www.iisermohali.ac.in/home/front-page-articles/admissions-to-int-phd-program-august-2018
IISER, Pune has the best research faculty among IISER institutes. Meanwhile, there is also an interview to judge your core interest and research ability in the respective field of Science which you want to pursue.

Yes, you can do this course abroad. You are eligible.
Good luck.

Yes, you are eligible for the same.
Good luck.

IHM Chennai. Should ofcourse be your first choice. They have a well trained set of faculty. A decent enough campus and facilities.

With this rank, you should try for physical counselling of NIT-Kurukshetra, NIT-Hamirpur, YMCA, DCRUST-Murthal and UIET-Kurukshetra.

It is available on their website and past rank records. Still, you may check with your marks in College finder sites like here in Shiksha.com.

Distance Education is the easiest to pursue. It will not matter which state are you in. You will only have to go the nearest exam center for exams.

IIEST-Shibpur beats all in terms of placement. Look at the figures and the companies who visit. Microsoft, Google, Pwc, coal India, Tata steel, Hindalco, ZS associates and what not. Be assured of a good future if you join IIEST.

The admissions are open, you can apply.