Some of your questions are not entirely clear to me but let's give it a go, shall we ?
For a horn balance on an elevator, where the pilot is wanting to raise the nose, does it move downwards to create an upwards force?
It's bit more complicated than that, unfortunately, although you appear to have the basic idea.
The horn provides two benefits -
(a) it can reduce the inertial elevator loads felt by the pilot through the controls (which makes for a "nicer" tactile feel to the elevator motions, and
(b) it is useful in reducing the push/pull loads which the pilot needs to apply to effect elevator displacement from the faired, neutral position.
I suspect your question relates more to the latter consideration.
In this case, if the pilot needs to achieve a nose up pitching moment, to provide a nose up pitching motion, he/she needs to cause the elevator to deflect upwards, which results in a change to the airflow around the stab/elevator combination increasing the downwards aerodynamic force on the combination. In turn, this causes the tail to pitch down and the nose to pitch up. For this process, horn aerodynamics cause the pilot to feel a reduced load required to move the stick rearwards (we talk about a reduction in the elevator hinge moment required). When the elevator moves up, the aerodynamic load tries to push it back down and the pilot has to resist this by keeping a pull load on the stick. If you have an exposed horn sticking a bit out in front of the elevator hinge line, then the airflow around the horn will provide a down force on the horn which, in turn, helps push the main part of the elevator (sitting behind the hinge line) upwards which reduces the overall load on the stick which the pilot has to resist by his/her pull load.
Does elevator trim up move up to create a local lift force, or to move the CP forward?
Answering this question would require knowledge of precisely just what you are after and, also, a detailed appreciation of any given trim mechanical design arrangement.
As a general observation, trim input is intended to vary the elevator hinge moment and, in turn, the fore/aft stick loads felt by the pilot, who has to resist any part of the moment imbalance which is necessary to hold an off-trim speed condition. This is why we have trim in the first place - the pilot needs to be able to get rid of any residual stick load needed to maintain an on-speed condition.
So, the trim mechanism provides a small, additional movement into the stab/elevator combination so that the overall aerodynamic loads can be varied sufficient to result in a (very near) zero elevator hinge moment to yield a zero stick load which means that the pilot doesn't have to continue actively providing push/pull loads into the elevator circuit. Achieving this will require a change in the pressure (and hence, lift) distribution over the stab/elevator combination. As with the flow characteristics, say, over the wing, the changes in lift distribution will necessarily result in a movement of the local CP to provide the desired change in elevator hinge moment.
How fast do wake vortices move on the ground, and they last in excess of three minutes, correct?
Shed vortex flow is a rather complicated subject. The vortices interact both with the ground surface and the local wind conditions. While we generally see the two vortices more or less moving outward a bit above the ground, they can rebound with ground and wind interactions and do rather strange things so far as their flight path might be concerned. Lateral velocities can increase significantly or, even, reverse in direction. Due to ground interactions, secondary vortices can be generated and these can interact with the original vortices. Furthermore, the initial vortex structure shape and energy will depend a lot on the specific generating aircraft's aerodynamic characteristics.
This makes it a bit difficult to come up with realistic, generalised answers to questions such as you have posed. Separation distances to in-trail aircraft become a tenuous matter: on the one hand the Regulators need to keep distances/times reasonably tight to minimise the adverse effect on airport capacity while, at the same time, keeping them sufficiently conservative to maintain a high probability that gust and roll interactions are kept reasonably within the structural and manoeuvring capability of following aircraft.
With the preceding in mind, notional answers might be in the vicinity of
(a) for decay characteristics, around 3 minutes for a reasonable level of vortex decay near the ground (decay tends to be at a greater rate near to the ground)
(b) for lateral nil wind speed, something proportional to wing loading / (aspect ratio x air density x aircraft speed), the equation coming from a NASA report. While the speed will vary, somewhat, depending on the characteristics of the generating aircraft, you could establish a rough order of magnitude in the region of a few knots.
more dihedral equals more lateral stability?
You would generally expect that sort of result.
Longitudinal stability would be greatest if CG was moved to the forward limit, or if it was move between midway and the forward limit?
Presuming you are concerned with static stability, the further forward, the higher the longitudinal static stability.
Max endurance and range would be highest in both super/turbo charged and normal aspiration engines flying low, due to an increase in airspeed at higher levels, is that true?
For endurance you are seeking minimum fuel flow. For a piston/prop combination this will be for low level and the engine settings somewhat appropriate for minimum power (as fuel flow follows power). Due to internal engine losses, with a constant speed unit you would be looking to set minimum RPM with the relevant MP for the required power setting.
For range, you want to achieve the minimum drag speed and maximum TAS (or GS if you are considering wind). End result is you fly at around full throttle height at the minimum drag speed.
Dynamic pressure is the square of IAS velocity, is that correct?
Close. Dynamic pressure is 0.5 x ρ x V2. Two options -
(a) local density and TAS
(b) ISA SL density and EAS. For light aircraft, we can use CAS in lieu of EAS. Given that the PEC is pretty small, we can use IAS in lieu of CAS with an attendant error. Better to apply PEC and use CAS.
