The most you can do within the claim is to delete a service code line with the "X" on its right.

Instead of using the Edit Claim window, use the Claim Details page. You will find the Close and Paid actions at the bottom of the Claim Details page and more. 

The blue bulk action keys are another way to mark many items as Closed or Paid.

To ensure that all the service code items are acted on in a claim, select Billing Cycle's Show Complete Claims search option.