You're absolutely correct, Boxer, and i forgot about the MCT thing: Yes, you have to briefly move the throttles out of the detent and then back into to get MCT after engine failure in FLEX.
The basic autopilot engagement mode is Speed mode with Vertical Speed as the vertical mode, and HDG. When you handfly around the sky and turn on the autopilot without flight directors on, it will engage in this basic mode. If a flight director was already on when you turn on the autopilot, it will engage in the current active modes.
For the GS/LOC thing: Our manual says not to arm the Approach above 8000' AGL (used to be 8200') due to, as Boxer said, lack of Radio Altimeter signal. If you still arm it, the system will revert to HDG and V/S upon Glide Slope intercept. Also, you have to have an ILS Approach selected from the database. If you select a non-ILS approach and push the ILS button, you get a flashing 'V/DEV' on the PFD. You always do something in the vertical direction (at least ALT or V/S up or down) and something in the horizontal plane (HDG ) So as a basic mode, the autopilot always engages in one lateral mode, which must be a HDG if the managed mode (NAV) is not active, and it engages in one vertical mode, and that must be either being level at a constant altitude (ALT) or going up or down with a certain V/S . The autothrottles engage either in SPEED when you fly V/S since this is a range of speeds so the autothrottles can adjust thrust to maintain this rate. Then there are the 'fixed' modes of CLB, MCT, TOGA which are fixed thrust settings. I think the basic idea is that when you turn the autoflight system on, it just keeps on flying what the airplane was doing at the time of engagement.
That's why it goes into Speed, V/S, HDG to just keep on doing what the airplane is doing this very moment. If a Flight director is on and you follow a managed guidance, then the autopilot engages in this current managed mode since it now knows what the airplane will be doing since it already knows where the plane will go, vertically and laterally.
For the sidestick: Pitch is a function of load factor demand. The system trims for 1 g level flight when the stick is in neutral. Whenever you move the stick in the pitch axis, you demand a loadfactor , up to the maximum allowed when you go to full deflection. If you yank the stick back, you immediately demand a high load factor and the bus will probably do something close to the limits for the current configuration (+2.5 to - 1.0 g for clean or slats extended and +2.0 to 0g for flaps extended).
Yes, configuration and bank angle for Vls. I've seen this, taking off with a heavy airplane with a turn toward the airfield after take off. When you accelerate to 'S' speed the Vls is not much of a factor ,but 'S' is very close to Vmax with the heavy weight, so you want to initiate the retraction right when you get to 'S' speed. If you are in a , let's say 20 degree or more ,bank, you might see Vls shoot right up through your current speed and the FOQA Swat Team (Flight Ops Quality Assurance) will storm your house the same night interrogating you as to why this happened.
I then usually fly at a slower speed until I can fly straight ahead and then accelerate to 'S' Speed and retract the Flaps/Slats.
G/S and LOC in the US are described in the AIM with the certified service areas. I think your standard LOC is good for intercept around 18 NM out. The question that came up: At ORD, Approach clears you to intercept the loc for let's say 14R about 45 NM out. Even though you get a good ID, you wonder: If there is a deviation out here tracking this LOC, am I then responsible for accuracy outside those 18NM?? No, some localiser facilities are certified to navigate at far greater distances. So it all depends.