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That's how it's usually translated in Spanish.

Cogito, ergo sum. -> Pienso, luego existo.

That's interesting, because staying truer to the latin would result in "Pienso, luego soy"

There's also an interesting confusion in Spanish, where "luego" can be understood as either "therefore" or "after"/"next thing". I think it's the same ambivalence you'd have with "I think, then I exist."

I don't know whether the latin original had that ambiguity.

Both translations from latin are indeed correct. But Descartes first used the french "Je pense, donc je suis" so yours is probably more correct.

Perhaps something along the lines of "My consciousness proves my existence" or "My awareness precedes my existence", depending on the interpretation.

I find this example harder to swallow:

"The question of whether to exist or not to exist comes to mind."

Descartes meant the latter, correct?

The former sounds much nicer.

Both. In this context, 'to be' is used in the same sense as 'to exist.' I'm not sure how you could use a different sense of 'to be' without rendering the sentence incomprehensible.