Droid is (or used to be) Motorola’s brand name for their Android phones. Google bought Motorola in 2012. They sold them later, but they may have retained the Trademark.
That cannot be right - Verizon (not Motorola) got the rights from Lucas Arts. IIRC, there was at least one non-Motorola Droid (made by HTC, IIRC)
Google did not, but Verizon Wireless did for the Motorola Droid in 2009.