to the forum!
I can't solve everything but I'd like to offer a few observations.
Whatever soil sprouted white mold all over it shortly after use should be avoided in the future. I don't know if that even affects the plants, but it's kind of a bad sign. So make a note of that. And clean your pots well after this episode, with plenty of soap or even bleach.
A huge variety of soils work for succulents indoors. At some point you need to learn how your soil behaves and adjust your watering accordingly. During the growth season, you want the soil to cycle from properly wet to mostly or fully dry. However long that takes, you wait. Then water well and repeat.
The most useful soils for cacti drain fast and dry out relatively quickly. This is primarily determined by the amount of rock in the soil. A soil mix of half organic, half inorganic (rock) is a good starting place. If the soil you buy comes with rock already in it, great. If it's mostly or all organic, add perlite or pumice (but not vermiculite or fine sand) to get it in that zone. And try to be consistent, generally.
Small pots will dry out faster than larger pots, as you have observed. This is because the water holding capacity is related to volume, but the rate of evaporation is related to the surface area of exposed soil. If you don't want a different watering schedule for every pot in your collection, you can just water the bigger pots half as often, simple as that, and that will nudge things in the right direction.
Mold and rot are treatable conditions (up to a point of course, and very much depending on the type) but they are far easier to prevent. That is where you will get the greatest benefit from focusing your attention and energy. Lots of light, lots of air flow, a good watering schedule where the soil goes properly dry at depth (not just at the surface, which dries out much sooner). These problems tend to take some time to set in, not just having the window open for a day.
I would recommend against unpotting and manipulating cactus roots unless you have a very specific question. You can figure out the moisture content in the soil without unpotting a plant. To whatever extent you may damage the roots by separating them from the soil and handling them, you may make the problem worse, not better. In any case, wait a week or more to water any plant that you bare-root or have otherwise broken roots.
