Q. Should I be getting all 3 deductions on each check?
A. Simple answer: Yes. Ideally, your employer would only withhold PA & Philly tax for the days you actually worked there. But, with an irregular work schedule, that's probably not gonna happen.
Q. Figured I’d be paying the PA tax and then either paying the difference or getting a refund when I file at the end of the year?
A. Yes, that's what will happen. You have to file a PA state return, regardless of the amount of withholding. If your employer withheld Philly tax correctly, you would not be required to file a city return. But, in your situation, you will have to file a city return for a refund of over withholding.
DE allows a credit for PA tax paid, but not the Philadelphia city tax. Reference: https://ttlc.intuit.com/community/state-taxes/discussion/can-i-get-a-philadelphia-income-tax-credit-...
