If I want to create an email address for every of my customers - do I have to host mail server or there is another way?
Sorry, I am complete new in terms of working with emails accounts. I know I can have postfix/dovecot/mysql or mailu or something like that running on my server but the maintenance and configuration alone puts me off even thinking about it. Is there some 3rd party service where I can create individual email accounts for my customers and access it using their API?
Hosting an email server is hard due to the many issues around SPAM and making sure your IP addresses don't get blacklisted, so if it's not your primary line of business and you don't have a lot of budget I would stay clear of it.
If you need POP/IMAP boxes for your customers, some email providers offer reselling options where you can provision whitelabeled account through their system, such as: https://mailbox.org/en/resellers
If you just need SMTP/sending and email forwarding, or you want to handle incoming emails with webhooks, check out https://improvmx.com/, https://www.mailgun.com/ or Amazon SES.