The tray icon can be stopped from running at startup. To do that you can open Spybot start center, checkmark advanced user mode, click Settings, then the System Integration tab, then click Stop beside Tray Icon. You can leave it installed or uninstall it, also.
Anything else running on startup would depend on which Spybot edition you have somewhat. Do you have Spybot Free Edition, or Professional, etc.? And is there anything running on start up currently that you particularly would like not to run, such as a scan or anything?