I think the car boots up first and you phone sinks to car. Then when carpuride boots up it switches from speaker to auto it will pause music when connecting. Dont think there is anything wrong. On my phone when I switch from bluetooth headset to speaker by video or music will do the same thing.

Use the aux cord. That will solve that issue.

Do you really need to have the phone connected to the Lexus HU at the same time ? If not disconnecting it might solve your problem.

What's it like in the sun?

just turn the car bluetooth off, problem solved.