You only need a regular PCR test taken max 72 hours before departure; the rapid test requirement only applies if arriving by air or sea
Show the test to Eurostar staff in London; no checks are made on arrival in the Netherlands to my knowledge
The check-in staff at JFK won't allow it. Not just for luggage reasons, but above all because, if anything goes awry with your connection, you're not "insured" and would be stuck in transit; unless you pay for a new connection, the inbound airline would end up having to take you back, a scenario they won't risk.
It depends. I don't know the specifics of the country you were denied entry to, but for the US, if you are denied entry at a port of entry in the US (all US ports of entry are in the US except for preclearance at certain Canadian and foreign airports), the officer has the choice of allowing you with voluntarily depart or to remove you. If the officer allows ...