I understand you have no sound on your system using 32bit Windows 7, but did with Windows 7 64bit. If this is not the case, let me know.
Were there any unknown/other devices listed in Device Manager?
I didn't see any Windows 7 drivers for this particular system. What happens if you install the following driver in compatibility mode?
1. Download the driver.
2. Run the download to let the extraction process finish, but cancel the actual installation. Take note of where it is extracted to.
3. Navigate to the extracted folder.
4. Inside the folder should be a setup.exe to install the driver with. Right-click this file and select properties.
5. Select the compatibility tab and enable compatibility mode for Windows Vista since it is a Vista driver.
6. Click apply/ok to save the setting. Run the file and follow the installation prompts.
Let me know what happens.