No, since you would need to enter into an agreement with Google for them to use your data, and they're unlikely to take you seriously unless you're affiliated with an actual transit agency. But if you're curious you can read about the steps involved.
Yes, it's possible, and there are open-source routing engines available for you to use, like OpenTripPlanner and Graphserver. This is pretty heavy-duty stuff, however. If what you have is a basic Web-hosting account and you just want to do "something interesting" with transit data, setting up an online trip planner is probably not the place to start.
I think the most straightforward solution would be for you to run OpenTripPlanner on a server of your own. This would provide your users with a familiar-looking website they can use to generate trip plans from your data while leaving you complete control over the data itself.
Note that running OpenTripPlanner would require a fairly powerful server plus map data from OpenStreetMap (which I'm assuming is available for your area) in addition to your own transit data. On the project's website you'll find setup instructions for Ubuntu to give you an idea of what's involved.
I'm assuming you're already able to generate a GTFS bundle; that is, to produce a ZIP file containing comma-separated data files as specified in the GTFS Reference. With an OpenTripPlanner server set up, your workflow would be as simple as
Making changes to your transit data.
Generating a new GTFS bundle.
Uploading the bundle to a specific folder on your OpenTripPlanner server.
Restarting OpenTripPlanner.
Optionally, notifying your users of the changes.
Every step except the first could be automated with a script.