Add type annotations to FeedController and related code (PP-3525)

[jonathangreen]

Description

Add comprehensive type hints to fix mypy errors in the feed controller and related modules, enabling strict type checking for these files.

Motivation and Context

These type annotations were noticed as missing while working on #2990. They are broken out into a separate PR to avoid making that PR larger.

How Has This Been Tested?

• Ran mypy and confirmed all errors are resolved
• All pre-commit hooks pass

Checklist

• I have updated the documentation accordingly.
• All new and existing tests passed.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 92.95%. Comparing base (3272ffd) to head (f4de643).

Additional details and impacted files

@@                          Coverage Diff                          @@
##           feature/suppressed-feed-improvements    #2992   +/-   ##
=====================================================================
  Coverage                                 92.94%   92.95%           
=====================================================================
  Files                                       453      453           
  Lines                                     43120    43120           
  Branches                                   6014     6014           
=====================================================================
+ Hits                                      40079    40081    +2     
+ Misses                                     1968     1967    -1     
+ Partials                                   1073     1072    -1     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.