1a: "She" is the definition. "either way" means it's a palindrome, and "mad" can be turned into the palindrome "madam", which can only refer to a woman - thus, "she".
6a: 'State" is the definition, and "I'm the slut" is a synonym of (in ebonics) "I da ho".
7a: "Will chill you right out" is the definition, since that's the point of Xanax. "This way or that" is another palindrome indicator. An executioner's weapon is an ax, and his target is a neck - so his target's head is "n". Using "ax", "n", the knowledge it's a palindrome and the definition of chilling you right out, a solver should end up with Xanax.
8a: "102" in Roman numerals is "CII" which is hidden in "Medicii". The Medicii family may or may not have spoken Latin, but at least they definitely had a presence in Rome!
9a: "Equipment" is the definition, since that's what Deere do. "Footloose" is both indicator and fodder: we want "E", which is the foot of "loose". Then we've got got "elder without large", so "Eder". I admit it's wrong of me to then also use "footloose" as an anagram indicator, but hey, I made this puzzle in fifteen minutes flat. "E" and "Eder" anagram to "Deere".
1d: "muddled" is both definition and anagram indicator - gain, rude and probably unfair. "Previous partner" is "ex", and "unlit" is "dim" both are read backwards in "mixed", so they return. Honestly a better clue would be "Confused return of previous partner in semi-darkness (5), which doesn't rely on the definition also being an indicator and uses has the necessary insertion indicator.
2d: Ads Lovelace is widely considered history's first programmer, and she can be found within the first, being 1a: Madam.
3d: A jig is a type of dance, and "dance" is indeed a word commonly used to indicate anagrams.
4d: This is my favourite stupidity in the puzzle. When asked what an elf would give a dwarf, Gimli - a "short man" - requested a single strand of Galadriel's hair. Thus, "a hair". "Ah" is a sound of fear, and an undressed fairy is "air".
Looks like Crosshare uses a version of markdown text formatting, so if you surround your text with ||vertical bars like this|| it'll be blacked out until it's clicked. (And yes, I had to do other fancy markdown to keep that from blacking out the vertical bars, what I actually typed there is \|\|vertical bars\|\|)