Is it a security issue if anyone can see a (SHA-512) hashed password of my user? How can I mitigate it?

That depends on your threat model. Sharing this hash with non-root users is generally less secure than not doing so, but you obviously knew that prior to asking this question.

What is the proper way to configure user's password via cloud-init?

That depends: which cloud / datasource are you using? Are you inserting user-data directly into this file? That's not recommended, though might 'work'.

Why is /etc/cloud/cloud.cfg read by all? Can I make it available only for root?

There are no secrets in this file, and cloud-init doesn't expect users to put these there. If a user puts secrets in a globally readable file, then yes, they have created a security problem.