Is it correct to assume that I can use the same primers, wads, powder and shot in either STSs, AA and Nitro and Gun Club? Does it make sense to just load and shoot a mix of the better quality hulls?
The answer to both questions is it depends on the powder. If you use a powder that is dense enough to allow the use of a "standard" wad in the AAHS, then the answer to both questions is "yes." But, I suspect using a powder that dense might lead to some dishing of the crimps in the Rem hulls. If you use a bulky powder, such as Clays, Red Dot, Promo (you can judge relative density by looking at the bushing charts, powders that need a large bushing for a given amount of powder are bulkier than those that need a smaller bushing for the same amount of powder), you will need a wad with a shorter crush section for the AAHS hulls. The recommended wad in that case is the DRA 12 wad from Downrange.
A "standard" wad is a wad that was not designed specially for use in the AAHS hull and, so far as I know, the only wad that meets this description is the DRA 12 wad.
In either case, you likely will not be able to commingle the two types of hulls while loading. You likely will need to make a loader adjustment at both the pre-crimp and final crimp dies when going from one to the other.
The AAHS is a great hull but the data from both Alliant and Hodgdon were developed using the AACF hull and while they say the data can be used for both types of hull, in many cases, the stack of the components is too tall for the amount of space in the hull and it could lead to buckling of the sides of the hull and/or a FUBAR crimp. When you see a load in the data that calls for am bulky powder and you are using the HS hull, there are 2 ways to cure the stack height problem. The first way is the use the DRA12 wad. The 2nd way is to use a wad of the same type (AA-type or Figure 8-type) but designed to hold 1/8 oz more shot than you are dropping. Such a wad will have a bit shorter crush section, allowing for a proper stack height. None of this matters is you use the same powders Win puts in the factory loads, WSH or WST. Both are very dense and the components called for the data give an appropriate stack height.