Since there are not many details about your code quality process, I will focus on your question:

Is it possible to somehow setup rules, to run branch pipeline only for MR and for default branch?

I think job rules may be something like this:

  rules:
    - if: $CI_PIPELINE_SOURCE == "merge_request_event"
    - if: $CI_COMMIT_BRANCH == $CI_DEFAULT_BRANCH
    - if: $CI_COMMIT_BRANCH && $CI_OPEN_MERGE_REQUESTS
      when: always
    - when: never

This set of rules uses GitLab environment variables like:

• CI_DEFAULT_BRANCH - true for your default branch (develop/main/master)
• CI_OPEN_MERGE_REQUESTS - true if pipeline has open merge request

Please experiment with them.