radio-archive: add /api/clip endpoint + download buttons + ffmpeg in Dockerfile

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

projects/radio-show/audio-processor/server/Dockerfile

@@ -1,5 +1,9 @@
 FROM python:3.12-slim
 
+RUN apt-get update \
+    && apt-get install -y --no-install-recommends ffmpeg \
+    && rm -rf /var/lib/apt/lists/*
+
 WORKDIR /app
 
 COPY requirements.txt /app/