I'd like to know how it is possible to invest in Tax Liens in Arizona and in other counties across the country by acquiring the property. How does one end up owning a property through Tax Liens in Arizona and in other States/counties?

And if this is a possibility, do you own the property free and clear once you do end up owning it?
Do you have to pay the delinquent homeowner's mortgage lender the difference if you sell the property?
Like if I end up owning it can I sell the place for its entire market value and get all that money for myself?

First, if there is a mortgage on the property its unlikley to go in a tax sale. The lienholder will pay the taxes to protect their interest.

Second, properties that go in a tax auction are often long abandoned properties in deteriorating condition that would require a significant expense to make saleable.

If they are in a reasonably good condition then the auction price will probably be a lot more than the taxes owed.