The Christian/Latin cross predates Christianity by far (to think, historically, that only Christianity established the cross would be foolish).
Now, to the question as to Tammuz being it's origin point - no, it did not originate with Tammuz. Why? Because Tammuz was not associated with a cross, but a taw/tav - a Hebrew letter that, at the time (being more Phoenician, being in Babylon's time), represented more of an "x" than a lower-case "T".
Here is an image of the characters of the Phoenician alphabet, showing the "tav" as an "x"
And this is an image of Tammuz, wearing a headband with the "tav" on it.
As we see in the above cases, the "tav" was a symbol represented with an "x", not a cross. Even Paleo-Hebrew, which was the form Jesus would have written in, used the "x" for tav until 135 CE.