Has picking and squeezing my mole made it cancerous or given me skin cancer? (photos)

Moles should not become irritated.  If there was no external trauma, the body's immune system may be reacting to it.  You should have it looked at by a dermatologist and consider having it removed and tested.

You cannot scratch or irritate a mole to cause it to become cancerous. It either has cancerous cells in it, or it does not. But if you find that you are irritating it, squeezing it, and scratching it, why don't you just have it removed to be more comfortable anyway? "This
answer has been solicited without seeing this patient and cannot be
held as true medical advice, but only opinion. Seek in-person treatment
with a trained medical professional for appropriate care."