could not activate cellular data network

What did your carrier say when you contacted them?

Try going into Settings>General>Reset>Reset Network Settings.

The problem is 100% related to your carrier. Cellular data is not provisioned on your account, or has been deactivated. Calling the carrier should resolve it.

Since you report just going from a foreign carrier, did you happen to install a carrier update to the phone when using this new carrier? If you go to Settings>General>About>Carrier. Does it display the current carrier that you have a SIM installed for, or the foreign carrier you were using? This could explain it, and restoring from the backup is putting that foreign carrier file back in. Try going to Settings>General>Reset>Reset All Settings. If that does not do it, then you are going to have to restore as new and sync your data back manually if your carrier cannot correct the issue. They should be able to push the carrier back to your phone if necessary.

Either via iTunes or iCloud, depending on what you sync your data to. I'm talking about wherever you sync your calendar, contacts, other data, and then music and apps through iTunes.

You can try restoring the backup. If that breaks it then there is no other option. You probably backed it up while on the other network, so the carrier settings were in the backup and get restored when you restore the backup. You should still be able to update the carrier settings with your "home" carrier; I'm surprised that didn't work. To do that (if you haven't already) go to Settings/General/About and tap on the carrier name.

That's quite unusual, a profile installed by a carrier. I'd be worried about what that profile did. For example, did it steal data from your phone, or monitor your communications? Profiles have access to data on the phone that apps and normal carrier settings do not.

I agree with Lawrence. What was the exact name of this profile? Supported carriers do not install profiles, just carrier updates. I'd be concerned as well.