Colleges in Roorkee

COER is the best option for B.Tech in Roorkee. It has good placements too.

First of all congratulations! I will recommend you to go with CSE from IIT Mandi, but in that case only if you have a keen interest in computer science and engineering. Placement wise it's very good also you will get very good exposure. IIT Mandi has very good semester exchange program too, you have a very good option there to go abroad and explore further. Else if you wish to go with core opt for the other options which are left.

IIT Roorkee offers admission on the basis of AAT (Architecture Aptitude Test).
AAT is an architecture entrance exam for admission into IITs. Moreover, for appearing in AAT you have to clear JEE Advanced also. To appear in JEE Advanced you should have cleared JEE Mains paper 1.

First of all congratulation for getting a decent rank in JEE Advanced.
Now you should decide which branch you want to pursue. If you are interested in civil then you should surely go for IITD. If interested in metallurgical branch, then go for IITB and if still wants to do mechanical, go for Roorkee.

No, you can't get admission in IIT's for M.Sc programmes. They are only available along with B.Sc programs as integrated courses of 5-years.

It's the best for students getting rank in between 6300 and 6800.
It's good in every aspect. You may also go for IIT, Mandi for civil at this rank.

IIT-Roorkee for engineering physics is the best option for you unless you have an extraordinary interest in naval architecture.