Does the Matter standard prevent devices from gathering information from other Matter devices, or even from your network/other networks around you? Does Matter dictate that devices may only share data with the Matter routers?

AFAIK yes - matter devices can theoretically collect information about other devices in your home. A lot of matter devices use very low power radio connections that would struggle to penetrate walls/etc and it works a little bit like the internet where a network packet has an almost infinite number of ways to potentially reach the destination. So, if you’re worried your TV might record the fact that your lights were turned off at 11pm… maybe get rid of that TV.

That is: are they prohibited from using non-Matter communication protocols?

Pretty sure Matter devices can use whatever non-matter communication protocols they want.

Can a device request that a Matter hub send data to a server on its behalf? Can a device directly talk to an external server?

Pretty sure they can, yes. And they need to be able to do that to download and apply security patches.