Is the serial number (of a MacBook Pro, OS X 10.7) ever sent to Apple? For instance, is it automatically sent during system updates or app store connections?
To my knowledge, each and every time your Mac will send a serial number to Apple, you are prompted to explain this and why it will be sent as well as whatever information is needed.
You can see this in action from the System Information app:
When you click the "Check my service..." item you will see this alert:
Similar notifications are presented when the App Store app needs to verify your specific Mac is entitled to Lion during a restore. The Console app (in the Utilities folder) can be configured to send your diagnostic information which includes the serial number as well as other information about the hard drive, software and error logs to Apple.
You can also ask Apple if they treat this information as personally identifiable from their main privacy page. From what I can tell, they go to great lengths to treat this as other sensitive information and go out of the way to not lump it with your personal account other than when you do that by asking for service from AppleCare or registering your Mac.
When you first go through the installation screens, the serial # can be paired with the Apple ID if you opt in to the registration and that information is presumably sent securely to Apple's servers for registration purposes.
Serial numbers, or any other of your identification information do not appear in crash logs or diagnostic information transmitted to Apple.