Vicks is what I use for the calluses on my feet. My feet crack and peel too. I went to the doctor and he took a sample scraping of the hard skin and said it wasn't a wart, wasn't fungus either. Just recommended I use vaseline and wear socks to bed. - But I found Vicks works better as it stops the itching associated with the cracked skin. I've been using Vicks on my feet for years now. Before I apply the Vicks, I slough off the dead skin using a porcelain nail file - guess you could use a pumice stone. - I did have a plantar wart years ago - but haven't had any problems since.
Just had my mom go to a podiatrist for the same thing. He told her it's because her feet are wet with sweat, to wear thinner socks and change them if her socks get wet. I had what doctors thought where warts on my shoulder, they tried compound-w and that made red lines travel up my arm. I then tried Tinactin anti-fungal cream and they went away and never came back.
You really have to visit a podiatrist. Plantar warts can be extremely painful and require professional treatment and evaluation. Ordinary warts are sometimes sensitive to soaking in an aspirin solution.
It looks a little bit like a verruca.
I suggest that every evening you soak your feet with some baby oil.
Pumice off the dry skin. You can also purchase battery operated little machines that take off the hard skin.
You can also purchase a cream from the pharmacist for hard skin and apply this to the skin on your feet. If the skin gets too thick it will start to crack and this is very sore.
For the verruca, you can get little corn pads and treat that, they can be very stubborn to get rid of. My husband has a patch of the same on his foot that he had had for years.
Best wishes.