Commercial DNA/Ancestry kits to confirm paternity?

Backstory: My paternal grandmother had five kids however, there is some speculation that she had an affair many years ago during her marriage which resulted in two children (one of them being my father). My father is deceased and the person who may have been his father is also deceased. The person who my paternal grandmother was married to and claims is the biological father is also deceased.
Commercial bought DNA kits (from Ancestry or 23andme) probably aren't the gold standard to assess these types of situations however, they're convenient (and relatively cheap for the realm of genetic testing) so it's likely my only option.

Questions:

1. Since I am the child of my father, I wanted to gain some feedback for whether there is anyone on my father's side of the family that I could compare my DNA with to prove the above theory?

2. The person who may have been my dad's biological father had children. If I shared a genetic link with those children, meaning their father is my biological grandfather, would genetic tests show more shared DNA segments?

3. My paternal grandmother's other children had kids - my cousins on my father's side of the family. If I compared my DNA to those people and we had different grandfathers, would there be less DNA segments shared?