feat: embed frontend into backend server with full-stack build pipeline (RCA-20)

The backend now serves the Vue admin UI as static files with SPA fallback,
eliminating the need for a separate web server. Dockerfile builds both
frontend and backend in a multi-stage pipeline. Added build:web and
build:all scripts, updated CI to verify frontend builds, and fixed
vitest config to exclude Playwright tests.

Co-Authored-By: Paperclip <noreply@paperclip.ing>

Dockerfile

@@ -1,3 +1,15 @@
+## Stage 1: Build frontend (Vue/Vite)
+FROM node:22-alpine AS web-builder
+
+WORKDIR /app/web
+
+COPY web/package.json web/package-lock.json ./
+RUN npm ci
+
+COPY web/ ./
+RUN npm run build
+
+## Stage 2: Build backend (TypeScript)
 FROM node:22-alpine AS builder
 
 WORKDIR /app
@@ -12,7 +24,7 @@ COPY tsconfig.json ./
 COPY src/ ./src/
 RUN npm run build
 
-# --- Production image ---
+## Stage 3: Production image
 FROM node:22-alpine
 
 WORKDIR /app
@@ -25,6 +37,7 @@ RUN npm ci --omit=dev --ignore-scripts=false && \
     rm -rf /root/.npm /tmp/*
 
 COPY --from=builder /app/dist ./dist
+COPY --from=web-builder /app/web/dist ./public
 
 ENV PORT=8080
 ENV HOST=0.0.0.0