I'm reasonably seasoned in Azure, though far, far from an expert. But it sounds like you need to set up a Reserved IP in Azure, that should solve your problem. You'll need PowerShell to do this. (tut:
https://azure.microsoft.com/en-us/blog/reserved-ip-addresses...)
I think you can do this from the Azure CP now, I am not 100% sure.
Your IP should not flush on reboot though, that should theoretically only occur if you're doing cold boots/shutdowns; but that's a no-no in any cloud environment unless you have a reserved IP; which is why you are encouraged to use cnames instead for domain management, etc.
As far as the region problem, this can happen if you're using something like a scale set or load balancers. That could throw off a trace route. Also making sure your storage account, etc are all in the same region is fairly important.