Recently I have a bad case of torn calluses while running The Giant.
Honestly, torn calluses from C&P is indicative of a technique issue. Cleans and presses are not (shouldn't be) particularly hard on the hands.
What do you think about using gloves for Kettlebell training?
Especially in my first few years of lifting KBs I experimented with all kinds of hand protection (tape, gloves, sock sleeves, gymnastic grips, etc.) and never found anything really satisfactory. Having any material between my hand and the bell makes holding on to the bell harder, and there was almost always slippage between the material and my hand which made gripping more difficult and just didn't feel good. Some people are philosophically opposed to wearing gloves or other hand protection, but my objections are strictly practical.
The key for me what just putting in a lot of reps over time, refining my technique, and listening to my hands. Stopping a session short because your hands are starting to feel hot is always better than getting a blister or tear that takes time to heal. But it's been a long time since I've had to stop a session short, and I can only remember getting a blister once in about the last decade.
Or do you have any advice in how to take care of your palm? The mistress starting to complain about mine.
Is the problem specifically tears, or calluses in general? Calluses themselves are unavoidable. You can file them down to keep them from getting too thick, but I find that as my technique has gotten better I rarely have to do this (I go weeks without needing to). For filing calluses I highly recommend a glass "nano" foot file (there are lots of brands on Amazon that all look the same). These files take the skin off easily without being too aggressive and leave a nice smooth surface, plus they are easy to wash clean. I find they work better on dry skin than wet.
The other thing is that you shouldn't grip the handle with your palm. You will get a callus line below the base of the fingers, but not actually on the palm. Some people do use a deeper palm grip for heavy snatches, but that's a more specialized technique, and after experimenting with it I tend not to use it under most circumstances.
P.S: I train in my living room, using chalk might dirty the floor.
I use chalk in my office at work with no problem. I keep a chalk ball in a Tupperware-style bowl and wipe off the excess over the bowl. You don't need a thick coating, just a light dusting. There's also liquid chalk.