I don't think this is at all guaranteed by free trade. You could have one country stuck in a rut where they're exporting only raw goods, and they have to buy everything from a (smaller) country making more advanced goods and selling them back. Of course, the balance has to even out, but it the money is spread between more people on one end then it doesn't mean they'll be as well off as the other country.
No, that's wealth equalizing. Raw goods moving from one country to another, means equalizing those raw materials across the globe. That is, those materials are not all concentrated in one country and forgone in another (inequality), but now spread more equally between the two countries. Same for advanced goods.
But you're not wrong, the real crux of where the inequality happens is in the bond between people - represented by money - which is nothing but an IOU, a promise of a favor. There's a reason why 2 nations are often leery about what currency to trade in, as it can easily be manipulated for unfair advantage.
I don't think currencies work the way you seem to be implying. While there are certainly economic advantages to being the currency everyone trades in (in particular for your ability to finance your way out of a recession with less impact on your currencies value), I don't think it is the source of any long term difference in economic strength, this has more to do (imo) with government policies and such. See e.g,. the wikipedia article on the resource curse.
Well no, economic strength is really a matter of labor productivity, and I do not disagree in that the difference stems from government policies, I am not sure what statement I said makes you think I disagree?
The OP questioned people's stances on wealth inequality and free trade, implying that the growing inequality stems from free trade policies. This is understandable from solely an American perspective, but globally and economically speaking, this is untrue. My argument is that it's rather government policies around credit (and its costs (ie taxes)) which fundamentally drive economic inequality.