Advanced Aviation India

For becoming a commercial pilot, there is a specific eligibility criteria, which you should first be aware of. This is decided on what subjects have you taken in class 12th. It requires PCM to be a pilot. And, even if you have not opted for the same, then you can appear separately for these subjects and become eligible. Further, you should be aware that you are medically fit to be a pilot, so I would suggest you visit http://www.dgca.nic.in/medical/medical-ind.htm and go to the class 2nd examiners tab to find out doctors who will examine you for initial fitness. You can visit any doctorate from the list as per your convenience. Once you are done with Class 2nd medical, find out a suitable flying club for you as per your convenience. You can fly in India or abroad. I would suggest you fly in India. Besides finding the club, be aware that you will have to appear for:
1. Air Navigation
2. Aviation Meteorology
3. Air Regulations
4. Aviation Technical
5. Aircraft Specific
6. Radio Telephony
Here, 1-5 are conducted by DGCA and 6 is conducted by WPC. Prepare for these subjects and appear, as exams are conducted every quarterly. For preparation, flying club will help you or there are private instructors available in metro cities as well. You will fly for 200Hrs + Clear DGCA and WPC exams + Clear Class 1st medical, then you shall get your CPL. The next license is ATPL, which you shall be able to apply for once. You will then have to probably fund or self-sponsor your type rating on whatever fleet you are selected in an airline. Then, you finally get a job as a pilot. I hope the process is well explained to you. CPL is an expensive affair. It might cost you around Rs.18-20 Lacs for flying + DGCA exams Rs.500/- each exam + Medicals for Rs.10, 000 + endorsements and license issue fees around Rs.25k + Type Rating around Rs.15 Lacs. So, decide wisely before you enter the field.

Please check for your eligibility to pursue a CPL course from the criteria mentioned below:
Commercial Pilot License (CPL);
A student can get the CPL by doing his flying training and complete 250 hours of flying.
Eligibility: Class XII (Physics and Maths). Current PPL License needed with flying experience in the last three years, 18 to 30 years and medical clearance from AFCME, New Delhi or IAM, Bangalore.
Written exam: A test in the following subjects is needed for getting the CPL: Air Regulations, Aviation Meteorology, Air Navigation, Technical, Planning. These exams are conducted 6 times a year. Actual flying experience required is 250 hours of total flying time (150 hours of solo, 25 hours of cross country, 10 hours of instrument flying, 5 hours of night flying).
After a CPL, you can build up on more flying hours, login 750 hours of flight time, clear the requisite papers and acquire a Senior Commercial Pilot's License. Regards.

It all depends upon whether you wish to make a career in military aviation or commercial aviation, besides the fact that you are medically fit and have passed Class 12 with Mathematics and Physics. In the case of Military Aviation (Indian Air Force / Indian Navy - Air Arm / Indian Army - Aviation Corp), you could choose the NDA (National Defence Academy) path towards fulfilling your dream in any of our 3 military forces. Alternatively, in case you pursue a Bachelors degree in Engineering or Physics or Mathematics, you have the option of joining the Indian Air Force or Indian Navy. For the Indian Army, one needs to be a university graduate. However, the degree can be from any stream. As far as commercial aviation is concerned, you can enrol for the Bachelor of Science (Aviation) degree programme from the University of Mumbai. It's a 3-year degree programme for career-oriented pilots. Con-currently, you will have to commence flying training, either in India or abroad (any ICAO contracting nation) towards your Commercial Pilot License and clear the DGCA Pilot License exams.

By course, I am assuming that you mean a major field of study, or curriculum and not a specific class. The answer to the second part of your question is a resounding YES.
Most commercial airlines require an associates degree at a minimum to get hired on with almost all of the major airlines requiring a bachelors degree. It doesn't matter if your degree is in underwater basket weaving, a degree is a degree and it checks off the box for the requirement.
Specialized degrees, such as aviation business degrees are a great bonus, but won't necessarily get you a job with an airline. They also limit you down the road if you have an epiphany and realize that you hate flying. Now you have to redo a bunch of course work that probably won't transfer to another field or degree. I would pursue what you enjoy in a field of study and see if there is a way you can tie it into aviation. Something like electronics engineering with a specialty or minor in aviation comes to mind.
The one specific thing that you can do to help to become an airline pilot is to start flying as soon as possible. As I have eluded to in other answers, while having a degree may be a mandate to getting hired on with the airlines, having the 1500 flight hours is also a requirement (there are some exeptions to this rule). You could take every college level aviation course out there, and if you don't have flight time, you will be stuck doing something else other than flying for sure. Whether you do the flight training in conjunction with your college course work, or attend a flying school not affiliated with your college is up to you, and both have their strengths and weaknesses.