Is double option mandatory In Europe and if yes where I can see in what countries it is?
I assume you are referring to double option in consent forms for different purposes. Well, if your organization is under GDPR, whether is based in EU or processes EU individual’s personal data, you will need to comply with GDPR worldwide.
To fulfill a form is always and in all the countries obligated to add the checkbox for marketing activities?
Yes, unless the marketing activity is the main purpose of data processing. I.e. in a subscription form to be informed about promotions. The purpose of processing is clearly marketing, and the user is giving consent for it. On the contrary, if the user is purchasing anything on the website or requiring general information if you want to use email for marketing purposes, you need to add the checkbox, because the user must be aware of what are you going to do with his/her e-mail.