Although at present there is no 'Single Market' for Crime and Punishment, is there mutual recognition of Criminal Laws across all EU members, such that any offense commited in one Country is also an offense in every EU Country? For example, a person commits a (non-corporate) crime in one country, and subsequently goes on holiday in another Country. Can he be prosecuted for the original crime (and possibly jailed) in the Country he is in while on holiday, without requiring extradition back to the original Country? Similary, lets assume that somebody that somebody publicly insults the EU (e.g. on Social Media) while in his home Country. Can he be punished anywhere in the EU for it, including his home Country?
The simple answer is no. This one of the areas least impacted by EU harmonization, the differences between judicial traditions are too large and there is little appetite for anybody to converge on this. As an example of an issue where the EU is nowhere near consensus is abortion: it is still illegal or almost completely illegal in some countries, completely decriminalized in other countries.
The first forays of the EU into this area of the law, under the guise of the “area of freedom, security and justice” are the European Arrest Warrant (technically not an extradition and in general much easier/faster) and the notion that some crimes should be punished everywhere in the EU (but the EU doesn't define the punishment, just that there should be one).
As you have noted, Justice systems in the European Union are withing the area of responsibility of the individual countries.
There are many reasons for this.
Constitutionaly, Justice Systems are a monopoly of the state.
Since the late 18th century, most (but not all) Justice systems in Europe have envolved from common law to civil law systems. Each of which have envolved differently.
There is no guarantee that every law (crime) defined in one country, will exist in another country.
A prosecuter appointed by one state, would be responsible for a geographicly specific area (up to the maximal area of country). International law, in general, does not allow prosecution in other areas of jurisdiction (violation of sovereignty).
is there mutual recognition of Criminal Laws across all EU members, such that any offense commited in one Country is also an offense in every EU Country?
Similary, lets assume that somebody that somebody publicly insults the EU (e.g. on Social Media) while in his home Country. Can he be punished anywhere in the EU for it, including his home Country?
If a person in Italy commited a Lèse-majesté and went to Germany, a German prosecuter could not make any charges because the corresponding law was abolished in Germany in 2018.
If a person in Germany commited a Lèse-majesté and went to Italy, a Italian prosecuter could not prosecute because
- it is not a crime where it was committed (Germany)
- it was committed outside their area of responsibility
If a person in Poland commited a Lèse-majesté and went to Italy, a Italian prosecuter could prosecute if requested by Poland, since it is a recongnize crime in both countries and the responsible prosecuter requested it.
The are many exceptions to these general rules, based on national laws
- France will prosecute when a French citizen is involved, independent to whether it is considered a crime where it was committed or (for whatever reason) not prosecuted.