737 will not autoland on one auto pilot. It will fly the GS into the ground. No flare at all. May look okay in the sim. not something to try in the aircraft.
Need to be dual channel to get "Flare armed" will not occur on one channel. No flare armed, no flare.
Aircraft is fail passive for auto approach. Ie it will not kill you if something goes wrong. It isn't fail active, where a failure will still leave the capability to autoland.
If one autopilot fails Both will disconnect. The reason for the nose up bias trim at approx 400 feet RA is so that when the disconnect, the aircraft will pitch up away from the ground. Will get your attention. Add power for a go around (or from auto throttle to maintain speed) and the aircraft really pitches up.
737 is also not approved for any single engine autoland. The auto thrust is not approved for single engine ops either. 737 autopilot doesn't control the rudder like 757 767 or A 320. So with an engine out it would yaw all over the place.
This pertains to -200 -300 and -400. Applies to SP 177 auto pilot. Early -200s with SP 77 auto pilot have no autoland capability.
I have never flown the next Gen 737. So can't speak about the capabilities of it.