Can you take the core Specialty of another class during getting better (the Specialty that you don't need to roll for) and is Fortune Fool effect only happen during character creation, or does it extend to every other hit point you gain later?
Yep! You can take any one Specialty when Getting Better, including one of the two core Specialties for a different Class.
Rules as written, Fortune's Fool can be taken later when you have more Hit Points to convert. However, it's possible this might be disruptive to normal play. I am very okay with GMs putting a cap on the number of Omens it can give, or ruling that it is available during character creation only.
As a Fool, you have to take Fortune's Fool during creation. I think I'm going to try and patch this, since currently it's better for non-Fools than for Fools.
For the Skinflint, Pinch Pennies causes your max Omens to change as your savings change. So if you gain $1,000, +1 Omens. If you then lose that $1,000, -1 Omens.