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/src/services/manifest_compiler.py

@@ -160,6 +160,11 @@ def compile_dockerfile(manifest: dict) -> str:
         lines.append(f"RUN groupadd -g {gid} {name} && \\")
         lines.append(f"    useradd -u {uid} -g {gid} {create_home}-s {shell} {name}")
         lines.append("")
+        # Set HOME and USER for runtime compatibility
+        home = f"/home/{name}"
+        lines.append(f"ENV HOME={home}")
+        lines.append(f"ENV USER={name}")
+        lines.append("")
 
     # Build scripts
     build_scripts = manifest.get("scripts", {}).get("build", [])
@@ -333,6 +338,21 @@ def resolve_mount_source(mount: dict, variables: dict[str, Any]) -> str:
     return ""
 
 
+def get_manifest_home_dir(manifest: dict) -> str:
+    """Get the home directory for a container based on manifest user config.
+
+    Args:
+        manifest: Fully resolved manifest JSON.
+
+    Returns:
+        Home directory path (e.g., /home/user or /root).
+    """
+    user = manifest.get("user")
+    if user and user.get("name"):
+        return f"/home/{user['name']}"
+    return "/root"
+
+
 def compute_image_tag(tool_name: str, manifest: dict) -> str:
     """Compute a deterministic image tag from manifest content.