Can I purchase an unsubsidized (month-to-month contract) iPhone 5 from Verizon and let my brother use it in Europe?
Yes, for Verizon phones in the US, neither the month-to-month contract (unsubsidized) phones (when they become available) nor the subsidized phones are in any way carrier-locked.

I tested this by establishing AT&amp;T service on my subsidized iPhone 5 before I did anything with the Verizon side of things and it has worked with both data and cellular provisioning on the nano-SIM from AT&amp;T.

The Apple LTE page will guide you to know which frequencies are available for which model of iPhone 5. The model of each phone is conveniently printed on the back plate of each device.

The carrier you intend to get service from is in the best position to guide you in picking a US model to import, but Apple's documentation show no difference in the GSM 3G capabilities of any of the three phones available at launch, namely the A1428 (GSM), A1429 (CDMA or GSM).