You can only pay online with debit cards if you use two factor authentication in Europe (eurozone). It's basically free though (cash is more expensive to handle), unlike credit cards.
No, the processor will redirect you to the page of your bank where you need to enter a code sent by sms by your bank. This is true 2fa, it works at least with all Russian banks I've used and I'm sure it works in EU the same way (I thought it was like that everywhere in the world?)
