Refactor Dockerfile, entrypoint script, and README for streamlined uv integration

- Added `UV_NO_CACHE` environment variable to Dockerfile for improved dependency handling.
- Updated entrypoint commands to use `uv run` for Python scripts and migrations.
- Enhanced README with detailed `uv` usage instructions, including installation, dependency management, and troubleshooting.
- Removed outdated `PATH` modifications for `uv` binaries in Dockerfile.

backend/Dockerfile

@@ -9,18 +9,17 @@ ENV PYTHONDONTWRITEBYTECODE=1 \
     PYTHONUNBUFFERED=1 \
     PYTHONPATH=/app \
     UV_COMPILE_BYTECODE=1 \
-    UV_LINK_MODE=copy
+    UV_LINK_MODE=copy \
+    UV_NO_CACHE=1
 
 # Install system dependencies and uv
 RUN apt-get update && \
     apt-get install -y --no-install-recommends gcc postgresql-client curl ca-certificates && \
     curl -LsSf https://astral.sh/uv/install.sh | sh && \
+    mv /root/.local/bin/uv* /usr/local/bin/ && \
     apt-get clean && \
     rm -rf /var/lib/apt/lists/*
 
-# Add uv to PATH
-ENV PATH="/root/.cargo/bin:$PATH"
-
 # Copy dependency files
 COPY pyproject.toml uv.lock ./
 
@@ -51,18 +50,17 @@ ENV PYTHONDONTWRITEBYTECODE=1 \
     PYTHONUNBUFFERED=1 \
     PYTHONPATH=/app \
     UV_COMPILE_BYTECODE=1 \
-    UV_LINK_MODE=copy
+    UV_LINK_MODE=copy \
+    UV_NO_CACHE=1
 
 # Install system dependencies and uv
 RUN apt-get update && \
     apt-get install -y --no-install-recommends postgresql-client curl ca-certificates && \
     curl -LsSf https://astral.sh/uv/install.sh | sh && \
+    mv /root/.local/bin/uv* /usr/local/bin/ && \
     apt-get clean && \
     rm -rf /var/lib/apt/lists/*
 
-# Add uv to PATH
-ENV PATH="/root/.cargo/bin:$PATH"
-
 # Copy dependency files
 COPY pyproject.toml uv.lock ./