Is hiking between Armenia-Georgia-Azerbaijan currently possible?

Are combined or cross-country walking holidays between these three countries possible at the moment?

It's technically possible, but due to ongoing diplomatic tensions, it's much harder to organise (and therefore more expensive) to organise a cross-border walking holiday between Armenia and Azerbaijan than normal. You'll probably need two separate tour operators for Armenia/Georgia and Azerbaijan, and travel in the walking border regions and remote villages is under strict control of the security services, making cross-border inn-to-inn walking holidays trickier than normal.