Will there be any constraints on functionality if I choose an Azure
region other than the Azure region pair (say, US East 2, US West)?
Based on my knowledge, there are no constraint on functionality if you choose the two location. US East and US East 2 are two data center and they are all in Virginia. Please refer to this blog:Tips for Choosing a Microsoft Azure Region.
If possible, you could select US East 2 and US West 2 as a pair. The two locations all provide G series VM. But for now, M series VM is not provided on US West and US West 2.
Also, based on my experience, G and M series VM will come to EAST US in the future.