How do you choose CAT II vs. CATIII ILS?
If both are available for the runway you're using, it's often easiest just to set up the Cat III, because you have the least risk that way of a random cloud between the Cat II DA and the Cat III DA interfering with your approach.

That said, there may be reasons to prefer the slightly higher mins of the Cat II, and those are typically specific to your equipment and your procedures.  Perhaps crosswinds, either steady or gusts, make the Cat II more desirable, or if your equipment has a limitation on "XYZ must be operative for Cat III approaches" and that happens to be deferred on MEL for you, then the Cat II is the better (only) option.  But generally, if both are available, setting up for the Cat III is usually preferred unless some specific factor drives you to the Cat II.

Separate from this, some runways only have a Cat II approach and not a Cat III available.  This is typically driven by runway lighting differences, although other equipment may drive this as well (specifically, a backup Glideslope transmitter, IIRC).

As much as I hesitate to answer "always," I can't think of any Cat IIIA DA that's other than 50' RA, and I can't think of a reason it would be, since you're over the runway itself by that point.  Cat II DA's can be slightly above or below 100' RA since the terrain you're over when you reach 100' above the TDZ (on glideslope) may be slightly above or below the runway elevation, but that's a Cat II thing, not a Cat IIIA deal.