Denti - how does your company permanent dual channel policy deal with disconnections below 400ft on landing? You are then manually flying an aircraft that's bloody miles out of trim which is hardly conducive to a stable approach.
Yup, due to this particular quirk of the 737 AFS it is out of trim. Dunno how they do it today, back when i got my typerating we had training on this. Including quite a few approaches in the SIM where we had to disconnect after the back trim was in and fly it to a stable landing. At least another one of those was later on done during line training. Although, my outfit allows FO CAT I autolands, which can help prevent the need for low level disconnection during CAT I conditions. Past 200ft (alert height) one needs a bloody huge problem to trip off the autopilot, in which case a go around might be a better option if still possible.
About the OEI case, this is what the manual (custom FCOM) in my company says:
Autopilot Engagement - One Engine Inoperative
When at a safe altitude above 400ft AGL with correct rudder pedal or trim input, the autopilot may be engaged. When VNAV is armed for takeoff, the autopilot is normally engaged anytime the airplane is above the minimum altitude for autopilot engagement.
In VNAV the AFS will usually command a shallow climb during flap retraction, but then, TOGA does, depending on thrust to weight ratio, command a 0-200 fpm climb as well. The only mentioned pitch modes for flap retraction in the S/E case are TOGA and VNAV with the latter one the preferred mode as it automatically uses the correct S/E acceleration height and has a LE protection speed feature.