I can't find BeyondPod widget

This is usually because BeyondPod has been installed on the SD card (e.g. it not installed in the main phone memory). Android automatically disables the widgets for any app that is installed on the SD card.

You can see/change where BeyondPod is installed by going to the global Device Settings, then select "Applications" (or Apps) and then selecting BeyondPod from the list of applications. If BeyondPod is installed on the SD card you will see a button with label "Move to phone". Select this button to move BeyondPod to the phone memory and your widgets should show up (sometimes the device needs to be restarted to see them).

The change above does not affect where your podcasts are/will be stored - just where the main BeyondPod executable is installed.