mammillaria compressa help

[CHADDYBIGGUMS]

I just got this cacti last night, it was given to me by a new friend. I noticed there's two spots that look like they are rotting right where the spines are, one of the holes is producing a milky substance now. Should I dab some rubbing alcohol on the effected areas or just let it try to heal its self? My second question is about the long taproot or corky bottom of the cactus. Can I trim off the old dead looking spines and re plant it down to the green part? I would usually just do these things without question but since this was a gift and it looks to have some age to it I really want to keep it in as good of health as possible.
Here are some pictures

[Aichybatoal]

Well, nearly all Mammillaria ooze a white substance when the skin is damaged. It's really sticky and nasty, but it's totally normal, and you should probably leave it there to do... whatever it does. It's not infected, so I see no reason to try and treat it.
Next, the brown, crusty bottom. I once had a (pretty deformed) Mam Mystax, and it had the same thing. Mine turned out to be hollow and rotten on the inside, and I tried to save it with a cutting, but it turned out to be rotten throughout and died. Anyway, it looked just like yours, and so I would check to see if it's just corking, or if it's totally rotten. Good luck!

[CHADDYBIGGUMS]

I ended up re potting it, it had a big long tap root if that's what you call it, and little roots coming off the bottom of that, it seemed pretty sturdy to me. Was it easy for you to tell that it was hollow once you had it out of the soil?

[Aichybatoal]

Well, a tap root is a root that emerges from the stem of a plant above ground, then burrows under from there. That cactus has no such apparatus - it's just the bottom portion of the plant. If it has small roots growing out of it, that means that it should be buried in the ground, because no cactus I've ever heard of has roots that intentionally grow above the soil. You know what I mean, right? Like, if there are roots growing out of that part, then that part should be underground, you know?

Anyway, it was very easy for me to see that mine had rotted. The brown area which I had before just considered age turned out to be practically a shell around mushy, necrotized tissue. If yours was rotting, you'd probably have realized it when you repotted it.