Hi Thomas,
Thanks for the in depth answer. The only problem remaining here is about the products related to Europe only - and this is why we came up with the .eu solution. Since we need to show a few products to Europe only, we can't use a single gTLD (.com).
Theoretically we could achieve this with geo IP targeting, but this option means that if someone from France is within the US and searching a specific product appearing only to Europe, he would not be able to see it - and we don't want that to happen.
The dev company working for this client made it clear that they can't and don't want to produce multiple directories for each countries (website.com/fr-fr/, website.com/fr-en/, website.com/de-en/, website.com/de-fr/, etc.). Same goes for subdomains (ca.website.com, fr.website.com, etc.).
Don't get me wrong, I'm not a huge fan of using two gTLDs (.com and .eu), but I don't see how we could resolve the European products problem without this option. I'm also worried about the sitemap, using two gTLDs. If you have an other solution in mind, it'd be more than welcome.