A) It depends on the aircraft spec. In our fleet there are both versions one that will not capture GS until LOC capture and other that will. Our SOP is that instead of having to remember which one does and does not all flights are flown in LOC first then after LOC capture APP is selected. If you still want to use APP only on the classic be very aware that you could end up with an aircraft that may head you down a false GS that may not be attached to a LOC.
B) My first instructor on B737 said that the rudder on the B737 has two uses, engine failure and foot rests. Whilst this was somewhat glib it was to point out that the rudder on the B737 is very powerful and should seldom needed. If you used rudder to correct LOC deviations in my airline you would not pass the check ride.