how can I find out which airports are operated by the FAA and which are FAA-contract (or even independently operated)? I'm currently looking specifically at IWA, Phoenix Gateway Tower.
The Air Traffic Control Tower at KIWA (Phoenix - Mesa Gateway Airport) is operated through Serco Management Services Company.
See the remarks section at Skyvector's airport information page:
ATCT OPERATED BY SERCO MGT SERVICES.
The FAA maintains a page with information about the FAA Contract Tower program, including this map (click for direct link):
Note that Non-Federal Contract Towers also exist; see this answer for more information.
Also NOTE: There are other questions on this site that have asked and answered similar questions. Some of the data (relating to FCTs and NFCTs) in the other questions are not current. The data in this answer are current as of the date of the answer.