feat: add FastAPI backend with PostgreSQL and start script fixes

- Add server/ directory with FastAPI backend
- Fix server/start.sh to properly handle venv on Windows/Git Bash
- Add alembic migrations and pyproject.toml
- Add server tests

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

server/alembic/env.py

@@ -0,0 +1,53 @@
+from __future__ import annotations
+
+from logging.config import fileConfig
+
+from alembic import context
+from sqlalchemy import engine_from_config, pool
+
+from app.core.config import get_settings
+from app.db.base import Base
+
+config = context.config
+
+if config.config_file_name is not None:
+    fileConfig(config.config_file_name)
+
+settings = get_settings()
+config.set_main_option("sqlalchemy.url", settings.resolved_database_url)
+
+target_metadata = Base.metadata
+
+
+def run_migrations_offline() -> None:
+    url = config.get_main_option("sqlalchemy.url")
+    context.configure(
+        url=url,
+        target_metadata=target_metadata,
+        literal_binds=True,
+        dialect_opts={"paramstyle": "named"},
+        compare_type=True,
+    )
+
+    with context.begin_transaction():
+        context.run_migrations()
+
+
+def run_migrations_online() -> None:
+    connectable = engine_from_config(
+        config.get_section(config.config_ini_section, {}),
+        prefix="sqlalchemy.",
+        poolclass=pool.NullPool,
+    )
+
+    with connectable.connect() as connection:
+        context.configure(connection=connection, target_metadata=target_metadata, compare_type=True)
+
+        with context.begin_transaction():
+            context.run_migrations()
+
+
+if context.is_offline_mode():
+    run_migrations_offline()
+else:
+    run_migrations_online()

server/alembic/script.py.mako

@@ -0,0 +1,26 @@
+"""${message}
+
+Revision ID: ${up_revision}
+Revises: ${down_revision | comma,n}
+Create Date: ${create_date}
+"""
+from __future__ import annotations
+
+from alembic import op
+import sqlalchemy as sa
+${imports if imports else ""}
+
+
+# revision identifiers, used by Alembic.
+revision = ${repr(up_revision)}
+down_revision = ${repr(down_revision)}
+branch_labels = ${repr(branch_labels)}
+depends_on = ${repr(depends_on)}
+
+
+def upgrade() -> None:
+    ${upgrades if upgrades else "pass"}
+
+
+def downgrade() -> None:
+    ${downgrades if downgrades else "pass"}