How to become a Theoretical Computer Scientist
You asked three questions:


Q1 is really out of scope for this forum: it relates to GATE and how you gain admission to Indian grad schools.
Q2 I interpret as "do I need a graduate degree to study theoretical computer science". In general, of course the answer is "no", but the discipline of studying at the grad level (with the access to professors and other grad students) is hard to replace with a self-study unless you're supremely motivated. 
Q3: of course you can apply to foreign universities for graduate studies: again, you might find it useful to peruse academia.stackexchange.com for a plethora of information on how to go about doing this.