Goin' to the Chapel

Question: Did Ishtar ask Gilgamesh to be her husband before or after she was married to Tammuz? Because going off the story order as written in your write-up, it was after she was married to Tammuz.

Yep, it sure was. In fact, Gilgamesh refers to her love affair with Tammuz while he's turning her down, telling her that marrying her is a bad idea for everyone and citing the example of how she's condemned Tammuz, the husband of her youth, to being stuck in the Underworld for half the year. In fact, check out his brutal assessment of her treatment of Tammuz on the sixth tablet of the Epic of Gilgamesh:

Of the beautiful shoulder and hand,
Tammuz, the lover of your earliest youth,
for him you have ordained lamentations year upon year!
You loved the colorful 'Little Shepherd' bird
and then hit him, breaking his wing, so
now he stands in the forest crying "My Wing"!...
You loved the Shepherd, the Master Herder,
who continually presented you with bread baked in embers,
and who daily slaughtered for you a kid.
Yet you struck him, and turned him into a wolf,
so his own shepherds now chase him
and his own dogs snap at his shins.

Dude is having none of it.

However, Ishtar being previously married to Tammuz doesn't necessarily mean that she couldn't marry Gilgamesh here as she suggests. For one thing, it's entirely possible that her marriage to Tammuz may not "count" during the winter months; he's dead, after all, and she therefore a widow and eligible to bang whomever she wants until he comes back and their marital connection is restored. In fact, despite the highly popular story of her romance with and marriage to Tammuz, Ishtar is also often said to be married to Anu, the great sky-god of the Anunna pantheon, apparently without there being any conflict (although in the Epic of Gilgamesh she's called Anu's daughter... but then again, that may be poetic, as her status as daughter of Sin is fairly well-established elsewhere in the mythology).

Perhaps most importantly, Ishtar is the goddess of love and sexytimes and fucking things up, so she probably has no problem whatsoever breaking rules about those things when it suits her. After all, the people who would try to stop her are fucking terrified of pissing her off, to the point where they actively enable her crusade against Gilgamesh; she's probably right to believe that nobody's going to stop her from marrying as many dudes as she wants.

It may also be worth noticing that when Ishtar says "marry me", she may also be speaking symbolically. She might not mean marry in the usual sense but rather be suggesting a sexytimes union of an impermanent nature, or even the priest-goddess sacred marriage that some Mesopotamian cults occasionally practiced, with Gilgamesh standing in as the priest thanks to his mortal blood. Or she might just mean "I want to bang". She's a mysterious lady.