[codex] Enable standalone web search in code mode (#26719)

## What

- Consume plaintext `output` from standalone search while retaining
optional `encrypted_output` parsing.
- Expose `web.run` to code mode and return search output to nested
JavaScript calls.
- Cover direct and code-mode standalone search paths with integration
tests.

## Why

`/v1/alpha/search` now returns plaintext output, which code mode needs
to consume standalone search results.

## Test plan

- `just test -p codex-api`
- `just test -p codex-web-search-extension`
- `just test -p codex-core code_mode_can_call_standalone_web_search`
- `just test -p codex-app-server
standalone_web_search_round_trips_output`

codex-rs/codex-api/src/search.rs

@@ -280,5 +280,6 @@ pub enum AllowedCaller {
 
 #[derive(Debug, Clone, Deserialize, PartialEq, Eq)]
 pub struct SearchResponse {
-    pub encrypted_output: String,
+    pub encrypted_output: Option<String>,
+    pub output: String,
 }