It's a public record so you should be able to get a copy. It may be available online.

Check with your local building department to see what their inspection protocol is. In our area, code compliance final inspections would include separate inspections for work performed by each trade (plumbing, mechanical, electrical and gas), a final building inspection (steps, handrails, tempered windows where required, etc.) and an environmental inspection (grading, drainage, and soil stabilization.) A (permanent) certificate of occupancy, when issued, means that all issues have been addressed.

I've never seen a building inspector write a report except in the case of a code violation which requires a written notice. A building inspector doesn't write a report like a home inspector; that's not his job. The building department isn't interested in taking on any more responsibility than necessary and it certainly doesn't want to relieve the contractor of his responsibility for the work.

The inspector gives the work a pass or fail so the most he might have on paper is a checklist. He often signs the back of the on-site permit card (usually in your window) for each inspection so take a look at it. If you pass all of the required inspections, you get a Certificate of Occupancy that gets put in the file for your address. The builder might get a copy to give you but it doesn't matter. You can call the inspector or go look at the file at the building department or online.

Yes, you should get a copy of the signed inspection card for each permit on your project and more than likely final payment to your contractor uses it as a definition.

Do not count on the building department's ability to keep archived copies.