GDP is a bit trickier than that, and exports are much harder to fabricate. Printing money clearly has its limits even for the USD (they can see the inflation now and the limits of it).

You'll need to export to a certain country, as far as I understand it. What countries usually do, is that they'll lend with their monopoly money while limiting the debit countries to purchase from them.

You can think of that as export-boosting but not completely fake exports.

What i've seen frequently is US investors buying EU companies that hold promise. So later it is a 'US' company and the narrative stays 'EU is great at basic research, but they suck at bringing it to market', but how can you beat free money?

I think you are right on this. Being the world's reserve currency, US can swap their printed money with real assets (in this case, companies, metals, grains, etc), thereby these companies bought with USD becoming American. If USD ceases being the world reserve currency, the dollar value can fall by 35% or more. In a way, the world (esp exporters, exporters who force importers to pay in USD) subsidizes US indirectly. Yet no one talks about this malaise, and the costs borne out by the the rest of the world. That's why the debates in US is always about why not print money to fiscal spending (for the ordinary people), as the cronies of capitalists have been printing money (or by suppressing interest rates, and by QE policies) to help the capitalist class and their elite servant class (managers, media, think tanks, etc).