I'm flying on Friday from the US to Luxembourg (via FRA). Our vacation (leisure) has us going from Luxembourg (4 nights) -> Belgium (6 nights) -> the Netherlands (5 nights) then home to the US from AMS. All EU travel is via train.

My understanding is that, as a vaccinated American, all I need is a CDC card to enter Luxembourg and there are no further requirements. Furthermore, I'm good to enter the Netherlands from Belgium with no testing requirements or entry forms.

Questions:
1) The Belgian website says Americans arriving in Belgium from the US need to provide a negative PCR and quarantine until an on arrival test is completed. Is that applicable to me since I'll be coming from Luxembourg where I will have already been for 5 full days?
2) Do I need to fill out the PLF for Belgium if entering from Luxembourg?
3) My understanding is that I can't get the covid QR code for the Netherlands since I'm an American. The thread discussing it here says the CDC card is good enough. Do I need a QR code for vaccination in Belgium? Can Americans obtain one?
4) It looks like Luxembourg also uses a vaccine passport ("Covidcheck")? Can I get this or is it only for EU citizens?