Including the H is the way the English decided to do it, "-tal" is from the German original. I'm drawing a blank on what "Google search drop" means.
Articles in English, of course, should use English spellings. IMO.
"Thal" is cognate with the English word "dale."
The German book Die Neandertaler by Friedemann Schrenk and Stephanie Mu+ller was translated into English and published as The Neanderthals.