Am I reading this correctly, that there is a fee for filing the federal return, a fee for filing the state return and an additional fee for processing?

You are posting from online Deluxe---there is a $59 fee for federal and a $59 fee for state; you are NOT required to pay the $40 refund processing fee if you pay for the federal and state fees upfront BEFORE you file using a credit or debit card.  If you choose to pay those fees from your federal refund, then you pay that extra $40 refund processing fee that sends your refund to a third party bank.  That bank receives your refund, takes out the $118 (+ sales tax) collects its own $40 from your refund and sends the rest to you.  Avoid that $40 fee by paying upfront.

How can I see my TurboTax  fees? 

  https://ttlc.intuit.com/questions/2565973-how-do-i-review-my-fees-in-turbotax-online

If your TurboTax fees are higher than expected, you can reduce them by removing add-ons     (BEFORE you e-file) :

• Remove Premium Services
• Remove MAX Defend & Restore
• Remove a state
• Remove PLUS Help & Support
• Remove Pay With Your Refund