Did the Joker kill Alicia Hunt?

I think in order to answer this, you have to carefully examine the character of the Joker and the way in which he phrases things. In addition, Alicia Hunt's mannerisms and the way she talks gives a big clue.

First, it's fairly well known that the Joker will lie to achieve his goals. He also has been shown in media to have trouble remembering how events unfold. And he's obviously a sociopath. Thinking back to the scene where the Joker meets with all the crimelords in Gotham (the following is pulled from here Batman scribed):

                            JOKER
              Nobody wants a war, Carmine.  If
              we can't do business, we shake
              hands... and that's it.

                            ROTELLI
              Yeah?

                            JOKER
              Yeah.

    ROTELLI stands up and extends a hand.  Joker shakes.
    A JOY BUZZER's concealed in Joker's palm.

    40,000 VOLTS course through ROTELLI's body.  He drops
    back into his seat a blackened husk, SMOKE pouring out
    from his sleeves and shirt collar.

This is a pretty good precursor to the Joker's behavior and sociopathic tendencies. Later, in a fit of rage at being ignored by Gotham in the newspaper, he poisoned household products and televised the murder of a poor man on live TV of how the chemicals will affect people. Later, in the scene at the art gallery, Vicki Vale is obviously scared and trying to distance herself from the Joker. She makes it clear what she thinks of him.

                          VICKI
              I think you're insane.

                            JOKER
              I am?  I thought I was a Pisces!
                     (brightly)
              C'mon, let's make up.  Here's a
              flower.

    There's a BRIGHT PURPLE BOUTONNIERE in his lapel.  He
    holds it up for Vicki's inspection as he moves menacingly
    closer.

                            VICKI
              NO!

    The JOKER squeezes a concealed BULB.  A JET OF CLEAR
    LIQUID spurts out of the FLOWER, NARROWLY MISSING VICKI.

    She GASPS, BUMPS INTO A TABLE.  ACRID BLACK SMOKE rises
    from the floor where the clear liquid hit.  Acid.

We know the Joker is infatuated with Vicki Vale, yet he is literally willing to put her life in danger. He doesn't seem to take kindly to being called insane here.

If we look at the parallels, Alicia wouldn't react much differently in the same position. The moment she sees The Joker she faints earlier on in the movie. In the next scene we see her, she is wearing the mask, asking where Jack is going. But the way in which she asks is clearly abnormal. It's safe to assume she is drugged, mostly likely self medicated. In the museum scene, we see she's been completely disfigured on one side of her face.

Having sort of established what type of person the Joker is and the way in which he reacts to people who reject him, it's probably safe to assume in a fit of rage he disfigured Alicia Hunt due to her not wanting to be with him anymore or some other form of fear.

Looking at the character description you linked of her. Down near the bottom, their definitely seems to be inspiration taken from the Black Mask character and his girlfriend.

Alicia was based on the character of Circe (the original Black Mask's ex-girlfriend and henchgirl). He also targeted Circe and forcibly disfigured her with the chemicals in order to force her to reunite with him. Circe would ultimately kill herself, leading Black Mask to replace her with a mannequin that he talked to as if it were a real person.

If we put all this together, it seems the Joker did in fact kill Alicia Hunt. Looking at the quote you provided it's obvious there is some rage in his voice when he reveals that you can't make an omelet without breaking some eggs. He then proceeds to smash the mask with his fist, almost as if he was finishing the very last bit of her he had left.

In speculation, it seems Alicia probably yelled at the Joker in a fit of clarity, with a combination of several emotions. One was most definitely resentment and great anger. She was also sad what he had done to her and most likely fearing for her life. And finally, he is pursuing Vicki Vale, which is making her feel abandoned and hopeless. Anything she might have said to him to show she was leaving or rejecting him, probably repeated events we have already seen The Joker do throughout the whole movie -- he killed her in a fit of rage by pushing her out the window. And just like Joker in several other mediums we have seen, he lied about the events or in his mind she actually did kill herself. But in reality, unlike the Black Mask story, the Joker was the one to commit the crime and to achieve his goals lied about the incident to make Vicki Vale feel sorry for him.

Yes.

While we don't know if the Joker physically threw her over, if we believe his claim she threw her self over because of his new outlook on life, he is still involved. He drove her to suicide. He scarred her. He scared her. He had violent tendencies and is very possessive.

She killed herself because of him, which would make him criminally liable for murder in a few states.

I saw that scene in 1989 at the premiere in my hometown along with my younger brother and even someone who saw it in California remembered the scene where he pushed her off a ledge and then the apartment scene came up. But it was not in the original VHS or even the DVD. I have tried finding the scene, but to no avail. I believe Jerry Hall had it removed because of her and Mick Jagger's children were young at the time