I want to put a passcode restriction on the Video app (and others), but the options I see in Settings are only for a few things. Is there a way, native or third party, to do this?
iOS only allows you to block built in applications and features, and age-rated apps from the App Store.
If you jailbreak, there is a tool for specifically locking apps individually called iProtect on the Cydia Store. However, it costs $8.99.
Apple provides an iPhone Configuration utility Mac application that allows installing various restrictions on an iOS device from the Mac. It's an Enterprise utility, but you don't need to be an enterprise to use it.