After touchdown with autoland, you must manually disconnect AP as 737 didn't came with auto roll-out ability.(you can stay it untill stop on 744,757-787 and buses.) normally you should click the AP disconnect button on yoke twice once to disconnect and once for silent the warning, which is available via SDK to blinding on your joystick or keyboard by lua, no UI for that then (I'd recommend use LINDA)
AT should automatically disconnected 2sec.after touchdown and should not set off warnings in this case.

Should you land manually, you need to disconnect AP above 50' as pre aircraft's manual, but our company set the rule for no lower than MDA/DA/DDA, and recommended to disconnect both AP and AT above 1000'AFE to get more time on hands.

If you want autoland, you have to engage both autopilot, after APP is armed and above 800'RA.
If only one AP engaged, the airplane should still flare IRL as I know, but it's rather a fail-safe design as single AP couldn't provide the necessary reliability required IRL. maybe not relevant in sim then.
Haven't try in CS's 737 how well it would do an autoland.

The 737 Classic does an autoland fine with both AP's on, but I haven't tried it with only one on yet.