The airline program is a phone call, the Metro is 15 minutes, and the dog question has an answer. Here it is, in order.
THE SHORT VERSION
Leaving the airport means entering Schengen, so the entry rules apply to you and to the dog. The Metro and trains run from Terminal 3. SAS arranges stopovers through customer service, so get it confirmed in writing. Big dogs usually fly in the hold, so ask the airline how a stopover changes the booking.
DRAFT
Every answer below is checked against the linked official source before this page goes live. Rules change; the "Last checked" date is the promise.
Can you leave Copenhagen Airport on a layover?
If you are allowed to enter Schengen, yes. The airport is about 8 km from the center, and the Metro (M2) and regional trains both leave from Terminal 3. Non-EU travelers register in the EU Entry/Exit System at their first Schengen border.
Our rule of thumb: six hours or more between flights makes a walk through Nyhavn and Strøget realistic, once you allow for the transfer and security on the way back. Under five, stay airside and eat well.
SAS lists Copenhagen as a stopover option, but it is arranged through customer service rather than a booking button. Ask before you pay, and get the stopover confirmed in writing.
Dogs travel on Danish public transport, and a large dog generally needs its own ticket. Check the current DSB and Metro rules before you go, because they are the ones the inspector reads.
Only if the dog meets EU entry rules for Denmark, because a stopover means entering the country. The Paw Passport guide walks through the paperwork in the right order.