I would like to let the polygons in my Index feature class set the extent of each page, I am sure this is possible, and the answer is simple, but I can't seem to find the instructions for doing this.
Data-driven pages will be centered on the features in your index layer, be they points, lines, or polygons. However, that doesn't affect the scale at which the pages are rendered. If you have polygons of varying sizes and you want your data-driven pages to have a varying scale to match, you will need to:
Determine the appropriate scale each page needs to be rendered at. Usually when I do this I'll try to limit it to 2-4 different scale levels and just know that every page isn't going to fit its polygon perfectly.
Add an integer field named 'SCALE' (the name doesn't matter) to your index feature layer and populate it with the scale value you want to use for each polygon (for example,
24000for a 1:24000 scale).
Click the radio button "Data Driven Scale" under the data driven page Extent tab (in your last screenshot) and choose your SCALE field as the field to use.
Then ArcGIS will render each page at the scale given in the SCALE field for each index polygon.
Warning: If you have a scale bar in your layout, it will change size as the map's scale changes from page to page.