Okay i've studied for an hundred of hours Canon law in university so i guess i can give an answer on that point. Even if your question, really required a book of almost 600 pages (you ask 2 things in your question).

First question the temporal status of papal states. It is a fucking hard question honestly. What we can say is, that papal claimed their land by Constantine Donation (that Constantine the emperor gave in his testement the western part of the empire to the Pope, the donation is a false clearly). This is thanks to this donation that they claimed to be able to crown emperors in the west, cause the just delegate what was their. This lead to theory of both glaives, the spiritual one and the temporal one. Glaive represent power here. The spiritual one is about "souls" and temporal one is about "living humans" Pope thoughts during middle middle ages that he had both power, but he was willing delegate the temporal one to the emperor. This lead to a crisis, known as investiture controversy between the Pope and the Emperor and this ultimatily lead to the war between Guelfe and Gibelin (both of those subjects are well documented litterally thousand of thousands of pages, and i can't just shorter it on reddit if you are interested just give a look).

Then i answer your second question. Since the Pope was the temporal leader of Papal states the canon law applied to those States. The status of Canon law (which apply to the inquisition and any court connected to the Pope) in comparison with other laws in Europe. The canon law as itself was waaayy ahead of its time, canon law is basically the procedure we used in our classical continental law nowadays. It was the first law to reject the trial by ordeal which was a huge advance by its time (in 855, in comparison France rejected it in 1258). In the canon law the conviction was individual, while in other law the conviction was a demonstration we will have to wait Beccaria and 1764 to obtain the same principle. You had the right to a lawyer, and witness which wasn't necessarly the case in Laicus court. The instruction was at charge and discharge, while in other laws the instruction was just at charge (logical since the conviction wasn't individual. Torture was only autorised under the accusation of crimen laesa majestatis divinae by the bullae at extirpanda in 1252 (while in most of other laws torture was used until the late 18th, but just this subject need also a 600 pages book). And it was canon law which institued in first prison instead of physical punishment. And death penalty wasn't possible in first sentence.

In France blasphemy under Holy Louis could lead to the lifting of your tong, that wasn't the case in Papal States. (even concerning religion fields Laicus courts were more severe)

So yes, by law regim, papacy totally affected it. It was way more wishable to be catch for any crime in Papal States or in any canon court including inquisition than any other courts. It was waaaayyy more soft.

Sorry for the resumé it is maybe incomplete on some part (or too quick), but it was the best i could in order to being readible on reddit. If you are really interested by the question, just read canon law books