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Would paying a dental bill with Chase FU for the points and then reinbursing yourself from your HSA account a good idea?
In addition, if I were to use my CareCredit for health expenses would I be able to use HSA to make direct payments to CareCredit or reinburse myself to pay them?
You should be able to reimburse yourself for any expense you pay that is eligible for HSA funds reimbursment. I would make sure I kept the original reciept. There is nothing that says how you have to pay the bill so using a points card is a good idea.
As far as paying Care Credit directly with HSA funds. I don't know for sure but I would not do it, especially if Care Credit let you charge something that is not HSA eligible.