What's your question? Ask

Use a knife and pry off the pups from the mother cactus.
Lay the cuttings in the shade so that the cut ends callus over. Bright sun will burn the pups.
The larger the cutting, the longer it will take to dry. When the end is dry (gently push your fingernail into the end, it should be hard)
The plant the pups into a well draining potting mix. Water but not over water to cause the pups to rot. Allow to dry between waterings.
Roots should form in 6 months.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/barrel-cactus/barrel-cactus-care.htm

It looks like you are in gardening zone 7, while barrel cacti are only winter-hardy in zones 9 and warmer. Therefore, I recommend bringing your cacti indoors for the winter.

I think your plan will most likely work, although there may still be some risk of the cactus being damaged by cold. Also, avoid watering your cactus in winter, because absorbing more water may cause it to become more susceptible to cold.

If your cactus does happen to be damaged by cold at some point, see the Frost Damage section here:
https://cactiguide.com/cactipests/#frost

Your Cactus may be suffering from root rot.

You can try lifting the cactus and repotting in a sterile cactus soil mix.
Here is a link with more information.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cactus-going-soft.htm

Yes, you can remove the cactus babies, known as "pups," and plant them, usually without harming the mother plant. These articles will explain how:

https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/removing-cactus-pups.htm
https://garden.org/nga/searchqa/answer/48157/

Your Cactus can only survive brief periods to freezing temperatures and is hardy to zones 6.

You are in zone 7a , so your cactus should be ok year round for winter. If the weather is particularly cold, you may want to add some protection with a garden blanket.