To clarify a few points:
1) As was said, trade companies are optional and you need to manually add provinces to the trade company to form/expand it. There are some very, very good reasons for doing this. As your provinces in India will be "distant overseas", minimum autonomy will be 75%, which means you get virtually no tax, production or manpower from them in the first place. Adding a trade company does a number of things: Doubles your own provincial trade power (which is good for you), increases the production of non-western countries in the region of the trade company (and thus increasing trade value, meaning more money for you) and finally, if you get over 51% of the provincial trade power in the trade node, gives you an extra, free merchant. Having all your provinces in a trade company also means that neighbours cannot westernise off those provinces. Unless you have a land connection to the provinces, I can't think of a single good reason not to form a trade company.
4) As was said, the idea group bonus for Expansion gives you a casus belli (Overseas Expansion) on all provinces with Indian/Chinese tech group (and Sub-Saharan too if you want to conquer East Africa). This is by far the best CB for taking India due to the significantly lowered aggressive expansion modifier. You could - if you have it - also use Holy War from the Deus Vult idea in Religious Ideas, but it's not as good. Still good, but not as good.
You may have taken the mission that gives you a claim on the entire Indian coastline, but as soon as you take a province there in a peace deal you complete the mission and lose your claims, so don't count on having that forever.
Is it worth taking an entire idea group to get a casus belli? That's up to you, but you might quickly find yourself with an Indian coalition if you just use conquest CB with very few claims.
6) Nope. They'll still get the same mission as you, and when they take it they get a -200 opinion modifier with you. If you look weak, they will press their claims to try and get a coastal province for themselves. And no, you can't just sell them one and be done with it, as they won't accept it due to having a different culture and religion. They will gladly conquer it though.
7) Probably. Unless you can some very serious trade power in the nodes in-between Goa and The Channel, you'll lose most of what you transfer to the other powers.
9) Just to add to Xara, production adds to trade value too, so building production buildings is not a bad idea either. But trade power/value first.
