Can the named personal representative of my father's estate cash two small checks that were issued in his name

This is a typical quandry. A.R.S. 14-1201 technically permits you to prepare an affidavit stating that the total estate is less than $50,000. However, in the case of banks, they have their own internal policies and procedures (determined by the home office back in New York, or wherever). So that method usually does not work for checks. Your next alternative is called a "summary administration." This is a simplified version of a probate. This works particularly well in your situation because you have a surviving spouse. The checks and other assets of the estate cannot exceed $37,000 in value. I suggest seeing an attorney to assist in this because the forms are not available online. (Compare this to the Affidavit for Collection of Personal Property, which is on the Maricopa County Superior Court website.)