This removes the last couple of doc references to the Window class in the framework, so that I can re-land the engine side change that removes the Window class (flutter/engine#22505).

* Move web integration tool tests to web.shard

Web integration tool tests depend on DDC changes in SDK. This change
moves them to a separate shard and subshard so CI bot configurations
can run them separately.

In particular, with will allow running those tests on dart CI flutter
HHH web bot instead of a non-web one, allowing early detection and easy
classification of issues caused by SDK changes as VM- or Web related.

* Enabled verbose mode for flaky web_tool_tests

* fixed local engine crash with --enable-asserts

* Disable failing tests, fix local engine with asserts

* Removed mac web_tool_tests bot

* Addressed CR comments

* Add disabled mac web_tool_tests and disable win web_tool_tests

* Added notificationListner sample code

* Fixed spacing

* fixed spacing

* Moved the variables into the Build method

* fixed typos

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* Update packages/flutter/lib/src/widgets/notification_listener.dart
Co-authored-by: John Ryan <ryjohn@google.com>

* [integration_test] Reland add a `run` method for proper reporting of test results

* Changes to path resolving from previous PR

This reverts commit 2a5aa294.

Additionally, fix a few NNBD issues.

Fixes https://github.com/flutter/flutter/issues/70408

This reverts commit 8749d955.

* Disable failing test due to SDK issue

* Disable failing debugger and hot reload windows tests

In principle this is backwards compatible; I just merged all the classes into one and deprecated everything that became redundant as a result.

Address comment in https://github.com/flutter/flutter/pull/66386/files#r522671103

* 03ff8a409 TBR: remove the extra values which was overshadowing the test result (flutter/engine#22483)

* e9b6a428f Roll Skia from a06b63c56ecd to 8ead30d51c86 (1 revision) (flutter/engine#22487)

* ee700513e Roll Fuchsia Linux SDK from fULjPqtx9... to B4PaMsNWM... (flutter/engine#22490)

* 4cbb684ef Roll Dart SDK from 3e502e0c7e04 to 41893ff76b0f (1 revision) (flutter/engine#22491)

* 38b6c22d5 PlatformViewIOS CreateExternalViewEmbedder refactor (flutter/engine#22353)

* bbcf19ad1 Roll Skia from 8ead30d51c86 to 011218edb590 (4 revisions) (flutter/engine#22493)

[flutter_tools] use initially parsed package config for language version, sound mode determination (#70323)

Revert "[integration_test] Add a `run` method for proper reporting of test results (#70075)" (#70466)

This reverts commit af5eb3b9.