Can a condo association foreclose on a property for back HOA rent out the property when the bank owns it and there is a mortgage on the unit?

Yes. This is an evolving area of law, but generally the HOA can get up to 9 months of back payments as a priority lien. Many HOAs are foreclosing now and the new owner at the foreclosure sale rents the property until the bank gets involved.

Most likely, but it depends on the law of the state where the property is located and what loans are on the property.

Read your HOA Bylaws. Generally, the association has a lien against the property for unpaid dues. Any such liens are subordinate ("junior") to the purchase money mortgage and any other existing liens. The association can then file a lawsuit to foreclose the lien and sell the property. Do you have equity in the home? Or is it underwater? There is no point in the HOA foreclosing the lien if there is no equity since the HOA can only collect sale proceeds that exceed the total of the senior liens.

Yes they can. They will however be responsible for the mortgage payments.

The HOA can lien the property for HOA dues and assessments. The HOA can foreclose on the property based on that lien.