If it helps you to advise others, whilst it may not be the same for all, I can give a brief overview of my experience:
In HK, they do not send the animals away from the check in desk until an hour and a half before the flight. He was terrified so the Cathay staff allowed me to stay with him by the desk and stroke him in the cage to keep him calm until he was taken. They were very good. In my case he was the only animal traveling, so if there were more it might be different, but they also offered that I could take him away from the desk and sit with him until the required time.
In Paris he came somewhat unceremoniously down the excess baggage belt which was nearest to the customs exit (Terminal 2E). There were no staff there to keep an eye or check on progress or welfare, so best to go to it as soon as possible. However it seems he came off the plane after the luggage, because my luggage was out before him.
When going through customs (there is no red or green there) I asked if I needed to declare the cat but they just waved me on - I guess they were relying on the checkin document checks.
Both myself and Sam were registered to travel with the eurotunnel in advance, but we still had to go to the pet reception centre to show all documents and scan the chip. There is a walk in entrance plus a drive through section so if needed you do not have to remove the animal from the van/car to process. This can apparently take some time, depending on how busy they are and how good the documents are, but we were through in a matter of minutes with all documents stated as in order and we could proceed to go through border control etc
The pet taxi company I used is called Pet Transport Services. https://pettransportservices.co.uk/
They are a small family run business and normally drive pets only but were happy to take me as well. They collect from the airport but do not do the receiving off the flight and they do scheduled trips but also bespoke such as mine. They were very informative and also responsive when my flight changed and I would recommend them.
Jodie is the lady that runs the office and can be contacted via [email protected] My cost was 1,500 GBP door to door but of course that may vary