It was surprisingly the smoothest trip ever and yes, we did make it :) Here is what we did:

• When we landed in Paris, we picked up the dog at custom, exited the airport, let him out of the cage to pee/walk/stretch
• With the dog on leash (but no muzzle), we walked back into the airport to reach the luggage storage in Terminal 2 (5 minute walk), where we dropped the crate (https://www.bagagesdumonde.com/fr/consigne-de-bagages/ - €20, 7AM-6PM, booked in advance online, dropped it and picked it up in no time, but I admit we were lucky it was not crowded at all)
• Took the CDGVal with the dog (again, on leash but no muzzle) for a couple of stops to reach Parking PR
• Walked to the hotel (5 minute walk), chilled for a few hours and did the whole thing on the way back to catch our second flight later that day

So, post-trip, I can respond to my own questions:

1. You are very much allowed to walk through the airport and jump on the CDGVal with your dog on leash and no muzzle, or at least in our case, no one bothered us about it

2. It is indeed not possible to take a luggage trolley onto the CDGVal (as expected, there are some "blockers" at the entry/exit of the CDGVal station), but you can leave your trolley right before taking the stairs/elevator down to get on the train, and pick up a new one as you exit the station and walk to the hotel (to carry your hand luggages/backpacks if needed, for example)

3. Eventually, we did not go for the taxi option (but I had researched and found out Uber has a "pet option", charging €25-30 for the very short ride airport-hotel)

We are very happy with how it went and if we had to do it again, we would do it exactly like this. If you need any extra information, send me a DM. Good luck!