I'd call customer service to check everything is setup ok and question the previous charges, although it's probably too late to do anything now.
Roaming guide here Guide: Roaming MegaGuide
With regard to your data. It sounds like a bolt on for the extra 20gb Guide: Where is my discount or Extra Data?
Worth a quick call again this morning @Class2ldn to remind them they promised a refund, esp if the bolt on was or should have been there.
If you are sure you have travel inclusive, yes, you will be safe to use your phone as normal.
For info, standard O2 travel will cost you £4.99 a day.
Travel inclusive is free use of your allowances.
Right now you will be getting charged.
You really need to call customer services to see why you don't have travel inclusive and if they can add it for you.
Track usage, upgrade and get support with the My O2 appDownload My O2