Many Syrian/Lebanese that ended up in the US in the beginning of the century may have spent some time in places like Cuba, Mexico, Dominican Republic, etc.David in Spanish is indeed "Da-vid" and I think your mother may have used the Spanish pronunciation if she that's the language they used at home.
David is a patronymic so it is likely that the real family name was not "translated".That's very common with Arab immigrants (seemy own name below...).
I would advise you to try and obtain US naturalization records for Thomas David and his wife.If they exist, you will surely find out a lot of information in there.
The confusion with Syria is very common as well.Lebanon only came into existence as an independent country after the French mandate.People who left before may have known their part ofthe world as "Syria".My own grandfather has documents that state that he was from "Mount Lebanon, Syria".It is likely that your ancestors were from what is present day Lebanon.And most likely Christians.
Finally, the brother may have been "Tannous" (Anthony).