Include examples in type checking

Remove `examples/` from the `pyrightconfig.json` ignore list and fix
the resulting type errors across all example files. Common fixes:

- Required API keys: `os.getenv("X")` -> `os.environ["X"]` so the
  return type is `str` rather than `str | None`, and misconfiguration
  fails fast.
- Narrow `LLMContextMessage` union members with `isinstance(..., dict)`
  before dict-style access.
- `assert isinstance(params.llm, ...)` before calling service-specific
  methods that aren't on the base `LLMService`.
- Guard optional frame fields (e.g. `LLMSearchResponseFrame.search_result`)
  before use.

examples/function-calling/function-calling-openrouter.py

@@ -60,11 +60,11 @@ transport_params = {
 async def run_bot(transport: BaseTransport, runner_args: RunnerArguments):
     logger.info(f"Starting bot")
 
-    stt = DeepgramSTTService(api_key=os.getenv("DEEPGRAM_API_KEY"))
+    stt = DeepgramSTTService(api_key=os.environ["DEEPGRAM_API_KEY"])
 
     tts = AzureTTSService(
-        api_key=os.getenv("AZURE_SPEECH_API_KEY"),
-        region=os.getenv("AZURE_SPEECH_REGION"),
+        api_key=os.environ["AZURE_SPEECH_API_KEY"],
+        region=os.environ["AZURE_SPEECH_REGION"],
         settings=AzureTTSService.Settings(
             voice="en-US-JennyNeural",
             language="en-US",
@@ -74,7 +74,7 @@ async def run_bot(transport: BaseTransport, runner_args: RunnerArguments):
     )
 
     llm = OpenRouterLLMService(
-        api_key=os.getenv("OPENROUTER_API_KEY"),
+        api_key=os.environ["OPENROUTER_API_KEY"],
         settings=OpenRouterLLMService.Settings(
             model="openai/gpt-4o-2024-11-20",
             system_instruction="You are a helpful assistant in a voice conversation. Your responses will be spoken aloud, so avoid emojis, bullet points, or other formatting that can't be spoken. Respond to what the user said in a creative, helpful, and brief way.",