Checkdocs checks issues, separate GitHub code

[kropidlowsky]

Description

Separated GitHub code for checking statuses of issues and added this functionality to Checkdocs.
Done some spot refactoring to Checkcomments.

Closes #4171.

Readiness checklist

• I added/updated unit tests (and they pass).
• I added/updated integration/compatibility tests (and they pass).
• I added/updated comments and checked rendering.
• I made spot refactorings.
• I updated user documentation.
• I ran task all, and it passed.
• I ensured that PR title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Milestone (Next), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 74.17%. Comparing base (fb4f384) to head (29a6461).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4258      +/-   ##
==========================================
- Coverage   75.27%   74.17%   -1.11%     
==========================================
  Files         323      323              
  Lines       22408    22408              
==========================================
- Hits        16868    16621     -247     
- Misses       4314     4576     +262     
+ Partials     1226     1211      -15     

see 31 files with indirect coverage changes

Flag	Coverage Δ
filter-false	?
filter-true	67.23% <ø> (-1.43%)	⬇️
hana-1	?
integration	67.23% <ø> (-1.49%)	⬇️
mongodb-1	5.11% <ø> (-0.01%)	⬇️
mysql-1	?
mysql-2	?
mysql-3	?
postgresql-1	43.10% <ø> (-8.81%)	⬇️
postgresql-2	41.70% <ø> (-11.76%)	⬇️
postgresql-3	43.29% <ø> (-10.20%)	⬇️
postgresql-4	40.79% <ø> (-0.03%)	⬇️
postgresql-5	45.83% <ø> (-0.02%)	⬇️
sqlite-1	42.35% <ø> (-8.63%)	⬇️
sqlite-2	40.96% <ø> (-11.66%)	⬇️
sqlite-3	42.66% <ø> (-9.95%)	⬇️
sqlite-4	40.06% <ø> (-0.05%)	⬇️
sqlite-5	44.74% <ø> (-0.08%)	⬇️
unit	31.88% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

[mergify]

@kropidlowsky this pull request has merge conflicts.