Oooh, you're so close...
The aetheryte ring cannot turn into any combat secondary stats. It's a terrible piece of gear to choose as an example, since it is an exemption to most of the rules. The aetheryte ring can just turn into all primary stats and elements.
This... isn't quite right. You will never get INT/MND/PIE materia from DoW gear, and you will never get STR/DEX from DoM gear. Additionally, if your gear is further limited, the list can get reduced even further (for example, if you're converting DEX-only job gear, then you can't get STR either which is the point that MeridaQ was trying to get across).
DEX doesn't affect tanks anymore, so it may not drop off of lv51+ tank gear (though I don't have enough data points to say that for certain), but I wouldn't be surprised if it was still in the materia-pool for tank gear from 1~50 just because DEX used to affect tanks.
All that said, what I really came in here to say is that I do not believe that the secondary stats on a piece of gear (crit/det/etc.) PROBABLY do not increase the rate of a piece of gear turning into those secondaries. Though I obviously cannot say that for certain, I don't think anyone else can say the opposite for certain either (if you have evidence to the contrary, I would love to see it). To answer the OP specifically, you should have an equal chance of getting det and/or crit from literally any piece of DoW/M gear.
