Add more handler tests

[w84thesun]

Description

Closes #2704.

Readiness checklist

• I added/updated unit tests.
• I added/updated integration/compatibility tests.
• 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), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

Merging #2769 (fc85f77) into main (8c818a4) will decrease coverage by 36.85%.
The diff coverage is 0.00%.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #2769       +/-   ##
===========================================
- Coverage   62.90%   26.06%   -36.85%     
===========================================
  Files         441      441               
  Lines       22727    22737       +10     
===========================================
- Hits        14297     5926     -8371     
- Misses       7499    16201     +8702     
+ Partials      931      610      -321     

Impacted Files	Coverage Δ
internal/handlers/common/setfreemonitoring.go	0.00% <0.00%> (-65.39%)	⬇️
internal/handlers/pg/msg_datasize.go	0.00% <0.00%> (-67.86%)	⬇️

... and 142 files with indirect coverage changes

Flag	Coverage Δ
integration	4.94% <0.00%> (-51.45%)	⬇️
mongodb	4.94% <0.00%> (-0.01%)	⬇️
pg	?
unit	24.53% <0.00%> (-0.06%)	⬇️

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[chilagrow]

Some comments 🤗

[rumyantseva]

I agree with Chi, it seems that errors returned because of invalid params should be tested and shouldn't be lazyerrors (even if they were before). But I like that we added more tests to check params!

[noisersup]

LGTM

[chilagrow]

Nice addition of tests!