Window framing gaps and split wood
Wow, where to begin...First, “adding” a window is “structural “ and therefore requires a Building Permit. The Code says the “owner of the property is responsible for obtaining all permits”. However, the Code says, “no work shall be completed until plans and specs are approved and a Permit issued.” 

So, your contractor could say it’s your responsibility to get the permit, but you could say he should not have started until it was issued and you’re going to apply for a permit next week. 

The bigger question is whether the “work” is acceptable. No, it’s not, because it does not meet, 1) minimum nailing requirements in Table R601.3(1) of the IRC, 2) framing requirements of Section R602, 3) no cripple stud (or nonexistent connection) under window sill framing, 4) questionable header size and lack of securing it to the wall. 

I see building paper directly fastened to studs, but it appears a plywood or OSB wall sheathing is installed on top of the building paper. If the house is covered in stucco, as you state, the stucco system has a moisture barrier and will trap moisture in the wall sheathing. I’d verify that I’m wrong...or cut out the paper on the studs. 

For you, building on an improperly installed base will only lead to future problems. I’d have it fixed before proceeding.