Hey Goldeneye, based on sightings we know that Paris was on 20 Nov and Almaty on 25 Nov. (Claude, you were right; I mixed up the dates for Almaty earlier!)
We see teams arriving at Berlin Hauptbahnhof at 3:50pm. Giving them some time to complete the tasks for Leg 6, they would have checked in from 7pm onwards. DeAngelo/Gary were released for Leg 7 at 1:30am. TAR rarely has <12h Pit Stops (except for keep-on-racing or penultimate legs) or >48h Pit Stops. This suggests that the Berlin leg took place on 22 Nov, and the Pit Stop was ~30h.
For Paris, we see Kaylynn/Haley checking in at 5:25pm (based on the clock on the Académie des Sciences) and checking out at 5:15am, so it was a ~36h Pit Stop.
20 Nov: Leg 5 and Pit Stop in Paris
21 Nov: Rest
22 Nov: Pit Start in Paris, train to Berlin, Leg 6 and Pit Stop in Berlin
23 Nov: Rest
24 Nov: Pit Start in Berlin, flight to Almaty
25 Nov: Leg 7 and Pit Stop in Almaty
Now working out some flight details from Berlin to Almaty, we see teams on two different Lufthansa flights: a narrow-body A321 (based on the aircraft safety card), and a wide-body with a 2-3-2 seat configuration towards the aft. In Lufthansa's current fleet, the only aircraft that fits this configuration are the A346, A343, and A333. Now Lufthansa doesn't use these aircraft for other possible connection points (e.g. Istanbul, Dubai, Moscow, St Petersburg), so the only remaining possibility is:
- LH 173, LH 45, LH 175, or LH 181 from Berlin to Frankfurt, 24 Nov morning, using the A321
- LH 646 from Frankfurt to Astana, 24 Nov 1320-2355, using the A333
- LH 646 continuing from Astana to Almaty, 25 Nov 0050-0245
I'm assuming teams were held at the airport until sunrise at around 8am.