WIP: Fix issues & merge requests badge counters sync

What does this MR do?

• Update badge count for issues and merge requests when issuable state is changed (closed, reopened), created, assignee changed.
• When issue assignee is changed from issues board.

Are there points in the code the reviewer needs to double check?

• I’m not sure about adding badge_count to the root path. I prefer to include it under a more intuitive namespace (maybe profile or api?).
• Did I miss some cases for expiring the badge count cache?

Why was this MR needed?

Screenshots (if relevant)

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Documentation created/updated
• API support added
• Tests
  • Added for this feature/bug
  • All builds are passing
• Conform by the merge request performance guides
• Conform by the style guides
• Branch has no merge conflicts with master (if it does - rebase it please)
• Squashed related commits together

What are the relevant issue numbers?

ref #32415 (moved)