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_git_mounts.py

@@ -148,7 +148,10 @@ class TestResolveSingleGitMount:
     async def test_missing_remote_url(self) -> None:
         """Git mount without remote_url returns empty list."""
         result = await _resolve_single_git_mount(
-            MagicMock(), {"mappings": [{"source_path": ".", "target_path": "/app"}]}, "/tmp", None
+            MagicMock(),
+            {"mappings": [{"source_path": ".", "target_path": "/app"}]},
+            "/tmp",
+            None,
         )
         assert result == []
 
@@ -157,7 +160,10 @@ class TestResolveSingleGitMount:
         """Git mount without instance_dir returns empty list."""
         result = await _resolve_single_git_mount(
             MagicMock(),
-            {"remote_url": "https://github.com/user/repo.git", "mappings": [{"source_path": ".", "target_path": "/app"}]},
+            {
+                "remote_url": "https://github.com/user/repo.git",
+                "mappings": [{"source_path": ".", "target_path": "/app"}],
+            },
             None,
             None,
         )