I strongly suspect that your best bet would be to get a taxi directly from your hotel to Seaworld. I don't know where the coach picks up, but Seaworld is near the bottom of Int Drive so you would probably go past it to get to the coach stop!

If you read through the various Florida threads on here you will find plenty of tips and advice. Any specific questions, ask away. We have always had a car so I don't know too much about doing it without one!

I would agree. As Disney want you to stay on site and certainly would not encourage you to visit a competitors theme park, I'm not sure where such a coach would leave from, if indeed such a service exists. A taxi will be the best bet.

If staying at a non-Disney hotel you will often find hotels offering free (or a minimal cost) shuttle service to the theme parks.

You could use Disney transport to get to Downtown Disney and get a cab from there.

Caribbean Beach is quite close to Downtown Disney so I doubt you would save much on the taxi fare but you would definitely add to the journey time. Just ask for a taxi at the hotel.