Initial extraction: groombook/api from groombook/app monorepo

Part of GRO-802 monorepo breakdown.

Changes:
- Extract apps/api/ as the main API service
- Inline packages/db/ (database schema, migrations, utilities)
- Inline packages/types/ (shared TypeScript types)
- Add CI workflow for lint, typecheck, test, build, docker
- Port Dockerfile with 4 stages: runner, migrate, seed, reset

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

Dockerfile

@@ -0,0 +1,47 @@
+FROM node:20-alpine AS base
+RUN corepack enable && corepack prepare pnpm@9.15.4 --activate
+WORKDIR /app
+
+FROM base AS deps
+COPY package.json pnpm-workspace.yaml pnpm-lock.yaml ./
+COPY apps/api/package.json apps/api/
+COPY packages/db/package.json packages/db/
+COPY packages/types/package.json packages/types/
+RUN pnpm install --frozen-lockfile
+
+FROM deps AS builder
+RUN mkdir -p /home/node/.cache/node/corepack
+COPY packages/ packages/
+COPY apps/api/ apps/api/
+RUN pnpm --filter @groombook/types build && \
+    pnpm --filter @groombook/db build && \
+    pnpm --filter @groombook/api build
+
+FROM node:20-alpine AS runner
+RUN corepack enable && corepack prepare pnpm@9.15.4 --activate
+WORKDIR /app
+ENV NODE_ENV=production
+
+COPY package.json pnpm-workspace.yaml pnpm-lock.yaml ./
+COPY --from=builder /app/apps/api/package.json apps/api/
+COPY --from=builder /app/apps/api/dist apps/api/dist
+COPY --from=builder /app/packages/db/package.json packages/db/
+COPY --from=builder /app/packages/db/dist packages/db/dist
+COPY --from=builder /app/packages/types/package.json packages/types/
+COPY --from=builder /app/packages/types/dist packages/types/dist
+RUN pnpm install --frozen-lockfile --prod
+
+EXPOSE 3000
+RUN apk add --no-cache curl
+HEALTHCHECK --interval=30s --timeout=5s --start-period=10s --retries=3 \
+  CMD curl -f http://localhost:3000/health || exit 1
+CMD ["node", "apps/api/dist/index.js"]
+
+FROM builder AS migrate
+CMD ["pnpm", "--filter", "@groombook/db", "db:migrate"]
+
+FROM builder AS seed
+CMD ["pnpm", "--filter", "@groombook/db", "db:seed"]
+
+FROM builder AS reset
+CMD ["pnpm", "--filter", "@groombook/db", "db:reset"]