A word up front: I don't really know anything about cacti, but I wish to learn!

I have one small & 4 very very small cacti. The small one and 2 of the very very small ones are exhibiting possible problems.

The whole family:

The small one on the left (Mammillaria?) was purchased around summer of 2015. It's one of those from Home Depot that come with the dumb flower glued on, which I very carefully removed at the time.

It's seemed to do ok until now, barring some rust-colored discoloration around the top (visible in photo below) that I first noticed maybe a couple of months ago.

It now seems there is a rot or fungus taking hold in one spot -- I've only just noticed it, but that area is fairly mushy. Upon very close inspection there may be a tiny bit of white spongy matter (too small to see in this pic, and different from the white "fuzz" in the crevices all over the cactus) in what appears to be the center of the affected area; radiating from that point is a dark region that I take to be dying tissue.

So there's that.

Meanwhile...

The four very very small ones were sent to me in mid-November, via an Amazon seller, by someone very special to me. These are them:

I have close-up photos of them from the day they arrived if those would be helpful, but have not included them here. All pics in this post are from this evening.

The issue here is that two of the very very small ones are exhibiting white patches they didn't have at first.

The white patches:

Now, some context:

In delivering the four very very small cacti, the UPS guy (or whoever it was) ignored the "this end up" sticker, resulting in soil spillage -- mainly in the two that are now affected. I put as much of the original soil as I could back into their pots, but also had to transfer a little more -- from the above-pictured larger cactus's pot -- to them.

Also, I am in central Alabama. The climate this time of year is all over the place -- a couple of days ago the high was in the 70s, but tonight's low is expected to be near-freezing. I keep all 5 cacti on a kitchen windowsill, the only place that gets decent (but this time of year not great) sunlight & is also out of reach of a toddler... The windowpane can get chilly on cold days/nights though.

The grow-light clamped to the sill is an inexpensive one that just arrived yesterday. It has the pink & blue bulbs, for a vivid purple light. It gives off almost no heat unless you wrap your fingers around the bulb housing, and even then it's only a bit warm to the touch. I thought it might help make up for the relative lack of sun this time of year. As I say though, it's brand new and therefore hasn't been part of the equation prior to yesterday.

As for watering, I've probably been doing too much... The larger one I've watered, only a bit, about once a week since I got it. The very very small ones I've waited to water until the pots were very lightweight and the soil "crunchy" if I slightly squeezed the sides of the pot; i.e., as dry as possible. I trickle a very small amount of water into the soil, not onto the cacti. From what I read this evening, it may be that they are dormant and I should not be watering them at all...

So:

Is surgery required for the larger cactus?

Do I need to apply anything (fungicide, vegetable oil, cinnamon) to any of these guys?

What about isolating any of them, and how far away from the others?

I've seen all these things mentioned, but again, I am essentially clueless.

And in general, how should I be watering? Might they be dormant now, or could the fluctuating climate be confusing them?

And is the grow-light helpful or harmful in these circumstances, keeping in mind that it is quite bright but gives off very little heat?

Sorry for asking so much in a single post! I love these little guys, and am eager to do whatever I can to help them.

Thanks so much for any life-saving advice, or even general tips, you might be able to share!