Penalty Shot Entry: On Penalty shot entry, after I click Penalty, the app is not letting me enter a distance.

For a penalty shot, the distance to the hole at the start of the shot isn’t needed, so the app makes it “easier” by not asking (nor allowing) you to enter that info.

In your example, shot 2 is a penalty, which was caused by the outcome of shot 1, so the strokes gained for shot 1 will suffer. The strokes gained for shot 1, when shot 2 is a penalty, depends on the beginning location of shot 3. So what matters is to enter the distance to the hole at the beginning of shot 3 (and then distance to the hole at the beginning of shot 2 is not needed).

Note: if you hit a tee shot out of bounds there is a “double penalty” button which automatically re-enters the shot 1 info for the beginning of shot 3.