Is it possible to restore a checked-out document using the Document Tool Restore? 

By product design, checked-out documents cannot be restored with the Document Tool Restore. 
Checked-out documents in the Web Client can be identified after performing a search with the symbol depicted below:

Alternatively, you can look in the XML header file in the storage location of the document under the Flags Node. Documents containing Flag Node "8194" would indicate a checked-out document:

KBA is applicable for On-premise Organizations ONLY.