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>

.gitignore

@@ -0,0 +1,23 @@
+node_modules/
+dist/
+.env
+.env.local
+*.local
+.DS_Store
+*.log
+.turbo/
+coverage/
+minimax-output/
+
+# Agent runtime artifacts — never commit
+.gh-token
+*.gh-token
+.config/gh/
+**/.config/gh/
+infra-repo
+infra-repo/
+**/instructions/.gh-token
+**/AGENT_HOME/**
+$AGENT_HOME/**
+.claude/
+.codex/