Brave search param changes and extra tests

[SergeyZhukovsky]

Resolves brave/brave-browser#15295
Resolves brave/brave-browser#15323

Submitter Checklist:

• I confirm that no security/privacy review is needed, or that I have requested one
• There is a ticket for my issue
• Used Github auto-closing keywords in the PR description above
• Wrote a good PR/commit description
• Added appropriate labels (QA/Yes or QA/No; release-notes/include or release-notes/exclude; OS/...) to the associated issue
• Checked the PR locally: npm run test -- brave_browser_tests, npm run test -- brave_unit_tests, npm run lint, npm run gn_check, npm run tslint
• Ran git rebase master (if needed)

Reviewer Checklist:

• A security review is not needed, or a link to one is included in the PR description
• New files have MPL-2.0 license header
• Adequate test coverage exists to prevent regressions
• Major classes, functions and non-trivial code blocks are well-commented
• Changes in component dependencies are properly reflected in gn
• Code follows the style guide
• Test plan is specified in PR before merging

After-merge Checklist:

• The associated issue milestone is set to the smallest version that the
  changes has landed on
• All relevant documentation has been updated, for instance:
  • https://github.com/brave/brave-browser/wiki/Deviations-from-Chromium-(features-we-disable-or-remove)
  • https://github.com/brave/brave-browser/wiki/Proxy-redirected-URLs
  • https://github.com/brave/brave-browser/wiki/Fingerprinting-Protections
  • https://github.com/brave/brave-browser/wiki/Brave%E2%80%99s-Use-of-Referral-Codes
  • https://github.com/brave/brave-browser/wiki/Custom-Headers
  • https://github.com/brave/brave-browser/wiki/Web-Compatibility-Exceptions-in-Brave
  • https://github.com/brave/brave-browser/wiki/QA-Guide
  • https://github.com/brave/brave-browser/wiki/P3A

Test Plan:

[SergeyZhukovsky]

post-init CI failed due to unrelated audit problem

[bridiver]

please either make the change now to use the test http server to handle the response or in a follow-up, but it is very important that we do not mock the response this way for browser tests

[SergeyZhukovsky]

new commits lgtm

[SergeyZhukovsky]

post-init failed due to unrelated audit errors

[tmancey]

Test failures are unrelated. Reached out to @emerick and @zenparsing

[SergeyZhukovsky]

macOS has 2 tests failures. I discussed it with @tmancey and it seems like it's a known problem related to rewards everywhere and not related to that PR

22:13:36  2 tests crashed:
22:13:36      BraveAdsBrowserTest/BraveAdsUpgradeBrowserTest.UpgradePath/PreferencesForVersion071WithRewardsAndAdsEnabled_ForSupportedLocale_ForUnsupportedLocale_RewardsShouldBeEnabled_AdsShouldBeEnabled (../../brave/components/brave_ads/browser/ads_service_browsertest.cc:1004)
22:13:36      BraveAdsBrowserTest/BraveAdsUpgradeBrowserTest.UpgradePath/PreferencesForVersion12WithRewardsAndAdsEnabled_ForSupportedLocale_ForNewlySupportedLocale_RewardsShouldBeEnabled_AdsShouldBeDisabled (../../brave/components/brave_ads/browser/ads_service_browsertest.cc:1004)