I am probably going to fly in through Paris with a 5 lb cat and 6 lb dog, the cat is a snubnose so she would fly in the cabin with me and the dog will go as checked/excess baggage. I inquired with Delta and the cost for excess baggage is $200.
I cannot bring both with me as carry-on, so I chose the cat
You could bring your 5 pound dog as a carry-on with no additional costs, according to Delta, just pay for your own flight, but keep in mind the carrier counts as your carry-on so you have to pack accordingly!
The maximum sized carrier (either soft side or hard) they quoted for me is 9 x 14 x 22 and it must fit under the seat and the pet stays in the carrier for the whole flight, you cannot take the pet out of the carrier.
Whatever airline you fly with, you will have to call them and ask for their requirements or if they even allow a pet as carry-on, as each airline is different.
As far as DEFRA, all the regulations are the same to enter France. Although you have to keep in mind that you have only 1-5 days after the tapeworm treatment done in the USA to enter the UK, so if you are spending a night or two in France and you go beyond the 5 days, you would have to find a vet and get the tapeworm treatment done again before you can enter the UK.
The Third Country Veterinary Certificate or the Pet Passport will be accepted in France and at Calais when they check the pet/paperwork prior to allowing you into the UK. But since the USA does not use the Pet Passport, those of us traveling from the USA use the Third Country Vet Cert. (Annex II)
Once you are in the UK, you can then eventually get a Pet Passport for your dog if you decide to take him out of the UK again.
I do not know what the procedures are for arriving in Amsterdam with a pet, I am ASSUMING they are the same, since the UK has since gone along with the rest of the EU pet regulations now? You would have to do some googling yourself to find out
Entering the UK would be the same DEFRA requirements along with the tapeworm treatment.
I have also been trying to find out if there are any custom fees or any other surprise costs for the pet(s) when arriving in Paris, but have had no luck, I am 95% sure there are none?? Also if there are any customs fees entering the UK via France, again, do not know???
I see this is your first post, welcome to the forum!