Copenhagen is very dog-friendly. Although a Nordic country, Denmark is more continental than Norway and Sweden and I found dogs were allowed in more places than at home in Sweden.
I googled 'dog-friendly Copenhagen' and found a few places that take dogs, including Sommersko og Cafe, Zoo Bar i Kronprisengade, Den Blå Hund, Optimist Cafe, Amalie
Cafe Retro on Knadbrøgade - but we didn't visit these, so I can't vouch for them.
We did eat at Cafe Norden and Cafe Europa, both on Amargertorv. Both are lovely, and Cafe Norden is particuarly famous for their desserts.
Plus, Nyhaven has outdoor seating year-round with heaters.