I can only see my call history from about 3 or 4 days ago.
You had to have expanded Call History to see the 3 or 4 days of items. It looks like Call History is expanded now, but no items are being displayed. Something changed. You aren't seeing the "Download" button because there's nothing to download. Cycle through different days selections and collapse and expand Call History to see if you can get items to display again. There's a known issue with Call History, so results may be somewhat unpredictable.