Can I walk in a grocery store in Denmark and after buying some stuff ask for withdraw cash by paying with credit card (i.e., as it's done in the US supermarkets)?
No, not with a credit card. Some stores do offer cash withdrawals if you pay for a purchase by card, but usually (or perhaps even without exception) only if you pay with the national debit card 'Dankort'.
As I understand it in most places cashback is only likely to be available with local debit cards.
Local Debit cards generally charge a flat fee per transaction so it's not a problem for the merchant to give cashback. Credit cards on the other hand usually charge a substantial percentage fee so merchants are unlikely to want to give cashback.
Debit cards aren't really standardised arround the world but the debit card issuers often work with a credit card network for international acceptance. So when you use your Debit card abroad it is likely to be treated as-if it was a credit card.