After takeoff, CLB is automatically engaged. Is this correct?
Yes, if the autothrust is armed.
Do you also have to *manually* engage the AP to engage NAV mode, or does it automatically engage at a certain altitude?
The autopilot has no influence on the NAV mode. NAV mode is a flight director function. It will engage automatically if armed at a preset altitude.
Then, in the climb phase, does the PF have to manually keep the AoA to whatever angle is proscribed, or is that then automatic as part of ALT or CLB (ie, at thrust reduction and accel altitudes is the pitch then automatic)?
Firstly there is no indication of angle of attack anywhere on the flight deck. You fly pitch attitudes. If you are manually flying then you follow the pitch attitudes indicated by the FD. If the autopilot is engaged you don't move the stick and the aircraft autopilot will control the pitch for you.
If the pitch has to be manually kept, surely that would disengage the any APs that are active?
Correct. Any movement of the sidestick whilst the autopilot is engaged activates the breakout and disengages the autopilot.