Double Option

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.

You can find more information in the following article:
What is the EU GDPR and why is it applicable to the whole world? https://advisera.com/eugdpracademy/knowledgebase/what-is-the-eu-gdpr-and-why-is-it-applicable-to-the-whole-world/  

If you have a question about to which countries is GDPR applicable, then please use this article instead: Is the GDPR applicable to our company? https://advisera.com/eugdpracademy/knowledgebase/who-needs-to-be-gdpr-compliant-an-easy-explanation/

You can also consider enrolling in this free EU GDPR Foundations Course: https://advisera.com/training/eu-gdpr-foundations-course//

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.

You can find more information here:
How does GDPR impact marketing activities? https://advisera.com/eugdpracademy/blog/2018/02/08/how-does-gdpr-impact-marketing-activities/
How does GDPR affect digital marketing? https://advisera.com/eugdpracademy/blog/2019/02/20/how-does-gdpr-affect-digital-marketing/
Email marketing in the era of GDPR – How to ensure compliance? https://advisera.com/eugdpracademy/blog/2019/05/27/gdpr-and-email-marketing-rules-for-compliant-campaigns/

Is consent needed? Six legal bases to process data according to GDPR: https://advisera.com/eugdpracademy/knowledgebase/is-consent-needed-six-legal-bases-to-process-data-according-to-gdpr/