Just bring the essentials. We pay for disposing of stuff here unless you are lucky enough that someone wants it when you give it away.

If you bring a bed from the US would that mean a different size making it a pain for sheets and a new matterass should you need one?

Diapers are one thing you shouldn't stock up on since they grow so quickly and they are in the next size before you know it. As for clothes, if you have a bigger baby, newborn clothes won't fit. I also didn't want my boy/girl dressed in neutral babygrows. The hospital provides the clothes for the first week and after that I loved choosing clothes for my little one.

As for buying clothes in bigger size for when they are older - if you can't be 100% certain for the gender then I wouldn't bother. Car seats may have different fittings too - Europe uses ISOFIX. Strollers are a lot cheaper in the US though!