Not sure what exact brand of pipe that is, but it looks like Dura Vent? What does the manufacturer install instructions say?
I don't think that type of pipe has the silicone gaskets, so the instructions probably say to seal the joints with silicone. Even if you had the gaskets, you would probably need to run a bead of silicone on the joints to make them stop leaking.
On the joint from the stove collar to the tee, you run a bead of silicone on the stove collar and then insert the tee/adapter over that, then apply 3 sheet metal screws (the self drilling type, small) to secure the tee/adapter on the stove exhaust collar (the thing here is to make the joint so secure that it can never separate on its own and set your house on fire). If ever you need to remove it, it is going to be a PITA. The important thing to do before you install the tee/adapter on the stove collar is to seal the gap between the double pipe with silicone, because that gap is prone to leaking smoke and it is easier to fill it with thick gobs of silicone before install. Use the orange, high heat silicone found in auto parts stores, RTV silicone. Or order the high heat black silicone if you want it to match the black pipe. Sometimes you can find it in the chimney/stove isle of the big box stores, next to the rope gaskets.
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For the joints between pipes with the twist lock mechanism, some people use high heat HVAC foil tape over the joint because it is easier to cut the tape to take it apart if need be.
As for the connection between the flex lines and that adapter, I am not sure, so I would see what the adapter manufacturer recommends. But I would guess it would be silicone bead and screws. I see what you mean about detaching for service. Maybe some one else will comment on that issue.
Good luck.