You can't buy FAA approved seats here but will need the European sticker on the seat. I flew with my baby in her infant seat last summer, we only got questioned on our connecting flight. She insisted on seeing the FAA label. I explained that it has the European label on it, she wanted to see it but couldn't find it right away. Turns out, it's on the very very bottom. In the end, she was happy with the seat. So make sure you know where your label is.
When we first moved here, I brought our son's car seat from the states, he was 18 months and since it reclines, he slept all the way over. The car seat was huge and it still fit fine. I happen to have a stroller that I could sit the seat on and then put my son in his car seat. It worked a treat.
I found on both trips, the airline staff is so helpful and they carried my seat or bags when needed.
Nobody at the gates even glanced at the seat. Certain airlines still book the eldery and parents with young children in first but if your airline doesn't, ask at the gate if you can get assistance if needed.
One thing that I have done on my flights is I took my children into the bathrooms just after getting off the plane to change nappies and freshen up. By the time we get to passport control, there are no lines. So much easier with wiggly ones when they don't have to wait in long lines.
WestHighlandWay, I would look at both airlines rules and would say that Delta would be the one to follow. Even though they are KLM planes, Delta is the company running them.