Is it acceptable to deny a POA that has specific gifting amounts listed since our bank cannot 100% monitor the activity? One example is "not to receive more than 5,000.00 or 5% of the value of my property in any calendar year" and the other is "in amounts not to exceed the federal gift tax annual exclusion at the time the gift or gifts are made".
Even if we place notes for the teller line, the AIF may still write checks and deposit at their accounts at other institutions and may have access to OLB and debit cards.

Most likely not your problem. If the POA grants banking powers - what they do with them in relationship with other limitations in the POA is not your concern. The POA most likely includes something like: I agree to indemnify the third party for any claims that arise against the third party because of reliance on this power of attorney.

Confirm with your legal counsel.