How does an auto-pilot control altitude changes?
Short answer:  it depends on which mode your flight guidance system is in.  It can be either number 2 or 3, although sometimes even something else entirely.  Whether you're using a target altitude or descent rate, it still all comes down to what mode you're in.  If you're in FPA, the elevator is adjusted to maintain descent slope.  If you're in VS mode, the elevator is adjusted for descent rate.  If you're in FLC/IAS/SPD, the elevator is adjusted for airspeed.  You should reference your pilot's operating handbook as the exact names and engagement methods for these modes varies a little.

When talking about autothrottle and constant speed altitude changes there are two independent inputs to the system, pitch and thrust, and two outputs of the control system, which are vertical speed and airspeed (or flight path angle and airspeed). You either control vertical speed with throttles and airspeed with the elevator, or airspeed with the throttles and vertical speed with the elevator.  It's possible to mix both throttle and elevator for both airspeed and descent rate (as indicated by number 1 and 4 on your list) but every flight guidance system I know of keeps the two separate to avoid confusion.

Advanced flight guidance systems often have three basic altitude change modes: flight level change (FLC), also sometimes called IAS flight guidance, vertical speed mode, and flight path angle mode. There's more on the differences between VS and FLC here.  I believe assume you're talking about the authrottle in airspeed control with the flight guidance in vertical speed mode.

In vertical speed with autothrottle speed control, attitude control and thrust control work somewhat independently .  Your elevator will be set based on the vertical speed control law to target a certain vertical speed, while your throttle will be trimmed based on its own control law to maintain the target airspeed.  So if you need to decrease descent rate but increase airspeed, your throttle will generally increase while your airplane will pitch up.    

You mention adjusting to the descent "slope", which would probably refer to flight path angle (FPA) mode. This mode basically tells the autopilot to adjust the elevator so that the flight path is at a certain angle (not the pitch, but the flight path).  This is a common alternative to vertical speed mode, and autothrottle will perform similarly for either mode.  If FPA is available the pilot can select it instead of VS on the guidance panel.  It's less used than VS, and you can read about why you would use one or the other here.

Flight level change (FLC), IAS, or even sometimes called SPD mode, behaves somewhat differently in that the elevator is used to determine your airspeed.  Thrust then determines your vertical speed and usually this is achieved by setting the throttles to climb or idle settings depending on the situation.

Of course, there's lots of what ifs that go into these systems like stall protection, overspeed protection, non-steady-state behavior, autopilot-authrottle interactions, and engine-out behavior.  I won't cover those details here.