Hi,
I'll attempt to answer your questions instead of simply giving you the instinctive response and support article link to you having mentioned the words "Apple ID" and "country". I hope you make it this far and read all responses to your question.
Two-region issue - I understand it is possible to set up a second Apple ID for my second region, and switch between them. However, can I do this without losing subscriptions, iTunes purchases, etc., and continue using my iCloud subscription in the original region?
First, I note that you are NOT talking about changing country settings on a single account, so a priori this document does not necessarily apply as written: Change your Apple ID country or region - Apple Support
Nevertheless some of the requirements will have bearing upon what you wish to do.
I very much doubt you can achieve what you want, particularly since you are using mobile devices which can only be signed into one account at a time, plus it involves subscriptions which will require you be signed in with that account in order for them to work. If you were just referring to items such as music tracks which you had purchased, those will remain on a device until removed regardless of the ID that that is signed in. Apple does not intend that a user have more than one account unless you need an entirely different persona that is completely separated from the first, plus Apple has things set up so an account is affiliated to a single region at a time.
There could be a bunch of other issues such as payment methods, though it sounds like you and your wife may be in different countries so that may be a way to arrange for that?
Also, is it possible to have family sharing across regions? (I figured I'd ask anyway :-)...) My wife and I have iPhones, and we share an iPad.
No.
Legal - Apple Media Services - Apple
"All Family members must share the same Home Country."
Share purchases with others in your Family Sharing group
Share purchases with others in your Family Sharing group on Mac - Apple Support
"To share purchases, family members must be in the same iTunes Store country or region."