is it true that the English version of Diego is Doug?

Answer 1

Diego is a Spanish version of James, another one being Iago. (Santiago and San Diego can both be translated "St James). So I would say that the English version was James, or "Jim" for short.
Douglas, on the other hand, is an anglicized form of the Gaelic name Dubhghlas, which meant "dark river" or "blood river" from Gaelic dubh "dark" and glais "water, river". Douglas was originally a river name, the site of a particularly bloody battle, which then became a Scottish surname. The surname belonged to a powerful line of Scottish earls.

Answer 2

I've always been told that Diego is the Spanish equivalent of James. But so is Jaime.
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Answer 3

No. It's David.