feat: expand ~ and $HOME in mount target paths

- Add expand_container_path() helper that resolves ~/ and $HOME/ prefixes
- Add get_manifest_home_dir() to compute /home/{user.name} or /root from manifest
- Set ENV HOME=... and ENV USER=... in generated Dockerfile for runtime compatibility
- Pass home_dir through instance creation and startup pipeline
- Expand mount targets in apply_resolved_profile() for regular profile mounts
- Expand mapping targets in _resolve_git_mount_mappings() for git mounts
- Expand working_directory and volume targets in _modify_compose_file()
- Update _prepare_manifest_instance to return home_dir alongside image tag
- Fetch tool_type early in start_instance to determine home_dir before profile application

Quality gates: pytest 188 passed, frontend typecheck clean

Addresses: home-path-expansion

apps/api/tests/unit/test_tool_instances_legacy.py

@@ -735,6 +735,7 @@ class TestStartInstanceManifestBranch:
             "headquarter/test:latest",
             "services:\n  app:\n    image: test",
             {"name": "test-manifest"},
+            "/root",
         )
 
         instance = ToolInstance(