If you can view your old conversation from a fresh installation on a new devices then this automatically implies that some 3rd party has access to your keys. I.e. your conversion cannot be considered truly private.
It can also imply syncing over an end to end encrypted (and verified, using QR codes at setup) channel between the devices being synchronised. I believe this is what signal does, for example.
No it doesn't. The sync could be device-to-device, or it could be encrypted in it's storage on the intervening server, and require that the user provide secrets on the new device.