Follow up, what were the punishments like for petty crimes like theft of food or being out past curfew, if there even was a curfew.

The world of French organized crime, called the Milieu (the middle) saw its heyday in the chaos and division of occupied France, especially in Paris. One thing to note is that a very many activities were politicised during this time, but not necessarily in the way you would think. First of all, the French vichy government supported Nazi Germany, meaning official state compliance with some of the more heinous acts of the Reich, such as jewish deportations. Police officers themselves were in some cases willingly assisting Gestapo, and many joined the french gestapo themselves (thousands estimated in Paris). On the other hand, there were hundreds or possibly thousands of officers secretly part of the Resistance. In Paris they formed three groups, l'honneur de la police, police et patrie, and front national de la police, and they launched strikes ahead of the allies and eventually fought during the liberation of Paris.

Criminals, especially from the Milieu, saw the advantages of the occupation. There are many well known Parisian gangsters who were part of the Gestapo, which gave them cover and backing while they organised prostitution, gambling, and black market activities. A well known example, Pierrot le Fou (Crazy Pierrot), joined for the immunity it provided his illegal activities. In 1944, with the tide turning, he would join the Resistance, making a name as an assassin. Whenthe war finally ended, he formed a gang from former gestapo and resistance members.

Although not a member, The french gestapo was also allegedly behind the protection pf notorious french serial killer Marcel Petiot, who lured and killed at least 63 would be refugees. He would later claim to have been a Resistance member fighting for France, his victims traitors and collaborators.

The most interesting criminal I've come across from this era and from Paris, Joseph Joinovici, managed to be part of the French Gestapo, an informant to the Abwehr, a major financier of the police resistance groupes and would frequently use his stolen money to bribe officials into releasing would be victims. All this while being jewish and born in what is now Moldova. He was only caught after the war, and his actions allowed him to escape a death sentence.

Besides in specific gestapo cases of hunting spies or enemies of the state, french police remained in charge of all other criminal proceedings, including in many cases against resistance members. Parisian police investigated what they saw as terrorist attacks against their city, and french forensic labs were in charge of conducting all analysis. For many of the pro occupation police forces, hunting terrorists and resistance members was a way for parisian police to prove themselves a reliable ally to the germans. And of course, Regular crime was still common in a major city like Paris, and the police remained in charge of most investigations.