In regards to Duty Free, I think it just concerns how much you buy. If you spend more than a certain amount, you will be taxed a fee when you declare what you bought. I think it's usually a large number per person though, so unless you spend ALOT it should be alright.
For this Duty Free, it's just what you bought in the country/on your travels.
I'm pretty sure you cannot bring any liquids/gels on your carry-on.
Duty Free really isn't THAT much better then department stores. Outlet stores/online are generally better.
I'm not positive on any of these answers, probably should wait for someone to confirm. but this is what I was under the impression of based on my experiences. Good luck, and have a safe trip!!