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_manifest_compiler.py

@@ -8,6 +8,7 @@ from src.services.manifest_compiler import (
     compile_entrypoint,
     compute_image_tag,
     deep_merge,
+    get_manifest_home_dir,
     merge_with_config,
     resolve_base,
 )
@@ -161,6 +162,38 @@ class TestCompileDockerfile:
         df = compile_dockerfile(manifest)
         assert 'CMD ["/bin/bash"]' in df
 
+    def test_sets_home_env_for_user(self) -> None:
+        manifest = {
+            "base_image": "ubuntu:24.04",
+            "name": "test",
+            "user": {"name": "dev", "uid": 1001, "gid": 1001},
+        }
+        df = compile_dockerfile(manifest)
+        assert "ENV HOME=/home/dev" in df
+        assert "ENV USER=dev" in df
+
+    def test_no_home_env_without_user(self) -> None:
+        manifest = {"base_image": "ubuntu:24.04", "name": "test"}
+        df = compile_dockerfile(manifest)
+        assert "ENV HOME=" not in df
+        assert "ENV USER=" not in df
+
+
+class TestGetManifestHomeDir:
+    """Tests for get_manifest_home_dir."""
+
+    def test_with_user_name(self) -> None:
+        manifest = {"user": {"name": "dev", "uid": 1001, "gid": 1001}}
+        assert get_manifest_home_dir(manifest) == "/home/dev"
+
+    def test_without_user(self) -> None:
+        manifest = {"base_image": "ubuntu:24.04"}
+        assert get_manifest_home_dir(manifest) == "/root"
+
+    def test_with_empty_user_name(self) -> None:
+        manifest = {"user": {"name": "", "uid": 1001, "gid": 1001}}
+        assert get_manifest_home_dir(manifest) == "/root"
+
 
 class TestCompileEntrypoint:
     """Tests for compile_entrypoint."""