Drag the Mail icon off the dock. Then you shouldn't see the bouncing. You might check in System Preferences, the Mail, Contacts & Calendars icon. Click on the Google account and deselect Mail if it is selected.
If you have no accounts in Mail, I can't understand why it would even be affected. If you deselected Mail in Notification Center, then alerts should not trigger anything in Mail to begin with.
However since you set alerts in a Google Calendar to sync with iCal to send email alerts which show up in Notification Center, I will assume that the OS is looking for an application to send the alerts to. That is probably why you are getting the Mail app activation telling you to set up an account to recieve the email. If this is the case, I wouldn't know how to stop it. Trashing the Mail application may create further problems because the OS won't be able to find it.
Maybe a different approach for you would be to set up the gmail account in Mail (set it up as IMAP). Once set up and working properly, quit the Mail application. That may solve you problem. Mail will have a valid account to recieve messages. You have to leave Mail activated though, but you can still deactivate the badges and sounds in Notification Center so it won't bother you. If you don't want to use the Mail application and it is set up properly as IMAP, you won't have to touch it anyway. It will stay synced with the webmail site.
I don't know what else to suggest except start your own discussion with your specific problem. Someone else may have the answer for you.
In addition: You posted in SL forum and you show you are using SL. I haven't used SL in years, but SL shouldn't even have Notification Center in it if I remember correctly. If you post a question, be sure to put it in the correct duscussion for either Lion or ML.