Yes, there are provincial offences, enacted by provincial legislatures. However, the criminal law, in the general sense, is a matter of exclusive federal jurisdiction. There is no provincial murder offence, or driving while impaired offence, or theft offence.
The provincial offences are regulatory in nature, and are only valid, constitutionally, if they are tied to the enforcement of a regulatory system. Highway traffic laws, for example, are provincial, and are backed up by provincial offences. Wildlife regulation is another area of provincial jurisdiction, as is securities law, so there are provincial offences relating to those topics. Provincial statutes often have specific offences, to ensure the statute is followed. Offences can include failure to file necessary documents as required by the statute, etc.
Criminal offences are governed by the Criminal Code, passed by Parliament, which includes criminal procedure. Offences can be prosecuted either summarily or by indictment, and in most cases the Crown prosecutor can elect which process to follow, depending on the severity of the case.
Provincial offences are passed by the provincial legislatures, and the procedure for prosecuting them is governed by provincial law. The procedure in each province is generally based on the summary procedure used in the Criminal Code, but with some tweaks for local conditions.
The constitutional requirements set out in the Charter of Rights and Freedoms for prosecution and trial of offences apply to federal offences and to provincial offences.
And to get back to the topic at hand: presumptive ineligibility is governed by federal law, and doesn’t take into account offences in other countries that don’t have an equivalent in the federal criminal law. A wildlife offence in the US, for example, will likely have no analogue in the Criminal Code, so wouldn’t trigger the presumptive ineligibility. But if the wildlife offence had an element of animal cruelty in it, that might trigger the presumption, since cruelty to animals is a federal criminal offence.