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-nvidia.py

@@ -60,17 +60,17 @@ 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 = CartesiaTTSService(
-        api_key=os.getenv("CARTESIA_API_KEY"),
+        api_key=os.environ["CARTESIA_API_KEY"],
         settings=CartesiaTTSService.Settings(
             voice="71a7ad14-091c-4e8e-a314-022ece01c121",  # British Reading Lady
         ),
     )
 
     llm = NvidiaLLMService(
-        api_key=os.getenv("NVIDIA_API_KEY"),
+        api_key=os.environ["NVIDIA_API_KEY"],
         settings=NvidiaLLMService.Settings(
             model="nvidia/llama-3.3-nemotron-super-49b-v1.5",
             # Recommended when turning thinking off