Add /api/health endpoint for uptime monitoring

[KonstantinMB]

The backend currently has no health-check endpoint. External monitoring (uptime checks, Render's own health probe, future load balancers) has to ping / or hit the FastAPI /docs page — both do more work than they should.

What to do

• Add GET /api/health in backend/main.py (near the top, not inside any auth-protected group)
• Returns 200 with JSON: { \"status\": \"ok\", \"version\": \"1.0.0\", \"database\": \"connected\" }
• DB check should be a single SELECT 1 — quick, no heavy queries
• If the DB is unreachable, return 503 with database: \"disconnected\"

Acceptance criteria

• curl http://localhost:8000/api/health returns 200 with the expected JSON when DB is up
• Returns 503 when DB is unreachable
• Endpoint is not rate-limited (monitoring tools poll frequently)
• No new imports beyond what's already in main.py
• Endpoint shows up in the auto-generated /docs page

Pointers

• Existing endpoint patterns: search main.py for @app.get near the top
• Database access: db is the global instance imported at the top of the file
• Version: pull from FastAPI(... version=\"1.0.0\") — line ~42 — or hardcode for now

Scope

Touches 1 file. Estimated time: 20-40 minutes.

Want to take this on? Comment below and I'll assign it to you.