Lets start by differentiating between the actual stall speed and the corresponding "certified" stall speed which is used to derive the minimum speeds required for the certification basis. The certified speed will be a conservative speed for any given configuration and flight condition. There are some significant differences in stall speed derivation of a prop vs jet powered airplane.
For a jet, the certified speed is obtained at idle thrust. Higher thrust will drive up the pitch attitude and at some level will result in a lesser weight being supported by the wings with a consequently lower stall speed. This assummes the jet exhaust is not impinging on a lifting surface (as on the C-17), which is a whole different doctoral thesis. On many prop powered airplanes the idle setting actually produces drag, so a setting "up off idle" is determined which produced zero thrust/drag, and it is used in determining the certified speed. Many prop powered airplanes have the prop wash impinging on the wing with resulting large decreases in stall speed as power increases.
Stall speeds do vary as one would expect with "high lift device" setting. On the sophisticated big transports these consist of the leading edge devices, trailing edge flaps and wing speed brake panels. Landing gear position can have an effect, but its usually very small. Anyboby with experience where the gear produced a big effect?
As you state stall speed increases as weight increases, usually in some straightforward relationship. On the big jets you can use the square root of the weight-ratio and be very close.
Center-of-gravity is also a variable. Certified stall speeds are determined at the most conservative CG position. generally this is at the Forward CG limit. The difference can easily be 6 knots from forward limit to aft limit on the big boys. This gets tricky when the forward CG limit is not constant but a function of weight.
Stall speed varys with Mach Number and thus varys with OAT and pressure altitude indirectly. On the smaller airplanes the difference in Mach number from max weight to min weight and from seal level to max certified altitiude is so small that the difference in stall speed can be ignored. On the big boys, with the difference in weight approaching 1/2 million lbs (max takeoff to min flying) and the range of altitudes from -1000ft to the mid 40,000's, the range in Mach number is large and the effect of stall speed substantial. The effect may be limited to the "clean" (ie everything retracted) config, and/or to just the extreme heavy weight regime.
The cert basis used prescribes the minimum ratios above the stall speed which must be used to define the AFM (or pilot handbook/manual) speeds. However, these are just one of the requirements which must be met in defining the minimum operational speeds. Typical other requirements are margin from stall warning onset, margin from natural buffeting (particularly during turns, and (recently) an ability to avoid a stall upon conducting an aggresive slowdown from the stated operational speed. Lastly the manufacturer may have his own requirements, based on his experience.
When the world worked with paper AFMS and Crew-Operating-Manuals these speeds were often simplified so as to provide the crew with rules of thumb (ie; during takeoff climb, flap retract speed is 25 knots above the V2+10 (all-engine climbout speed), and then LE-device-retract-speed is 35 knots above that. Boeing has had a long standing rule of thumb for landing speeds, which I don't recall now. Now with the powerful computers on board the big boys, these speeds are displayed to the crew real time and usually reflect all the nuances.
Hopes this helps.
