Hi bharris,
No, you do not need to stop any services, the installer will handle that for you. In general, we do recommend that you make a backup of your database in case anything goes wrong.
One caveat for the 5.3.3 to 5.6.2 upgrade: The installer will not let you proceed with the v5 key applied. For reasons I won't go into here, 5.6 and beyond use a higher version key. So before you run your upgrade, go to your Customer Portal, get your new key, deactivate your existing key on STM, run the upgrade and then apply your new key. This will not be the case for future upgrades, it was a onetime change for 5.6.
Regarding agent upgrades:
As of 5.6.2, the installer makes backup copies of any agent configuration XML files. These backups are not made via the "Server Push" method in 5.6.2 (but we are implementing it for a future release). Other than that, customers run their upgrades both ways, so it's really up to you.
Hope that helps!
