Design

Yes, you can definitely choose M.Des. after doing B.Tech in mechanical engineering. You can either go for industrial or transportation design, both are equally good in terms of placement and are relevant according to your degree.
There are only two colleges that have M.Des. in transport design:
1. MITID, Pune
2. UPES, Dehradun
Top colleges for M.Des. in industrial design are:
1. Lady Sri Ram College, DU Delhi
2. NIT, Bhubaneshwar
3. NIFT's
4. IIT, Roorkie
5. UPES, Dehradun

CATIA is a job oriented course. It will be a good idea if you complete CATIA, do a job and then pursue your MS or even MBA. If you get an year back, doing CATIA will help in improving your profile.

Yes you must have to clear the entrance exam for gaining undergraduate degree. many institutes have thier own individual exams like NIFT/NID ETC.

If you are already done with a web designing course. Assuming that you have done web development, you are far ahead of the crowd in the Indian market. Assuming you have done with some basics, you must be knowing JavaScript, HTML and CSS along with some backed programming language like PHP, python etc. Question yourself if you want to do web design and love it. Else, there are many better options.

Android is having a better market in terms of payments for Android application developers. Web designing lags much behind in pay scales. Also, Android is coded in java and hence allows you to work on a language used and accepted worldwide in IT. Android can also give you a chance to develop applications, being a freelancer.

Hi
This is a difficult question to answer. Starting a company needs idea, money, team to implement the idea and people, to buy your idea or product of idea. Web designing is not an only parameter to start a company. Yes, for now you can remain a freelancer who can develop web application and other related ones. Read some articles on how to bring up a startup, difficulties faced, etc.
Best wishes.