As a Canadian can I stay in a European country longer than my flight layover on the way to an Asian country?

E.g. Leaving the Frankfurt airport as a stopover and staying in the city for a week before resuming the rest of the flight to Asia?

Is it possible? YES
In Frankfurt Passport Control and Customs will decide if YOU can enter Germany. NOT YOU.

Canadians generally do not have issues entering Germany. Will YOU? Depends on YOUR situation.

Citizens of Canada DO NOT Need a Schengen Visa to enter Europe.

Are you a dual citizen?

There's a few issues.
The big one is if you miss your connecting flight, any tickets connected to the reservation including the return to Canada if you bought a roundtrip will be cancelled. Unless you purchased a refundable ticket, you will lose the tickets you have. Airlines do not allow you to do stopovers unless it was indicated at the time of purchase and booked with the dates. You can't simply miss your flight and come back a week later and expect to use the ticket. Won't happen.
I'm not sure how it works in Germany, but if you plan on entering the country, you may have to provide documentation on how you're leaving and your same day ticket would not qualify depending on the actual layover time.
You should call your airline and see of you can add the stopover. There will likely be a significant fee

The problem is that if you fail to board your connecting flight in Frankfurt the rest of your flight reservation will be cancelled. If you wish to stay in Frankfurt for a week you can make a "Multi City" reservation. You can reserve the flight to Frankfurt for January 1, for example, (you will arrive on January 2) and the onward flight for January 9, all on the same itinerary. This will give you a week to visit Frankfurt.

As a Canadian citizen you should not have a problem entering any European Union country without a visa for up to 90 days, as long as you have a return or an onward ticket.

No idea. Why not look at the German immigration website for information.