Do backups and uploads and other things continue running when I press the power button or when the computer display goes black to preserve power?
It depends. You can sleep just the display, or you can sleep the entire machine. If you sleep just the display, the computer is still running fully awake. If you press the power button, the computer goes into sleep mode so that most (but not all) operations are suspended to save power.
In recent versions of OS X, it is a bit more ambiguous, because the PowerNap feature means more types of operations may occur after you sleep the Mac, including Time Machine backups. For example, from this Apple document OS X: About Power Nap:
When your compatible Mac goes to sleep, Power Nap still works to do the following:
- Mail - Receive new messages.
- Contacts - Your Contacts update with any changes you may have made on another device.
- Calendar - Receive new invitations and calendar updates.
- Reminders - Reminders updates with any changes you may have made on another device.
- Notes - Notes updates with any changes you may have made on another device.
- Documents in your iCloud account - iCloud pushes any edits you made to a document to your Mac notebook.
- Photo Stream - Your Photo Stream updates with new photos from your iPhone, iPad, or iPod touch.
- Find My Mac - Locate a lost Mac notebook even when it’s sleeping.
- VPN on demand - Corporate email updates securely.
- Mobile Device Management - Remotely lock and wipe the computer.
Power Nap does more when your Mac is plugged in to an AC outlet:
- Downloads software updates
- Backs up with Time Machine <<<
- Performs Spotlight indexing
- Continues background downloads of Mac App Store items, including software updates
- Updates Help Center content
- Wake on Wireless support for Apple and third party wireless base stations
So if your Mac is sleeping and not plugged in, it may not back up. But if it sleeping and plugged in, it may back up, if it is compatible with PowerNap and PowerNap is switched on.