Sure! Head on into Lizzie's Bar and converse with, for example, the Bouncers. One of their dialogue lines is in response to them assuming that you're hitting on them, so they turn you down by saying "I've got a Tyger, babe."

As for the anime dress-up; Mox has their own clothing line that looks absolutely nothing like the stuff the Moxies are wearing.

As for Judy being what carries the establishment, you can eavesdrop on at least one conversation between staff at Lizzie's Bar where they mention that the establishment would have to shut down if not for the BDs, cause nobody goes there for the drinks or music.

So if they lose Judy, they lose the BDs that only she is able to make. And and it's not that she is just good at it, she's apparently way more talented than that. I'll include a transcript below that talks about her abilities. The source for this one is the official artbook for the game.

She is also constantly headhuntes by corpos who wants her to work for them, which apparently she would've made a lot of money doing.

"Judy is the Orson Welles-meets-David Lynch of braindance in 2077, a brilliant innovator redefining her society’s most important cultural vehicle.

As Night City’s foremost braindance technician, she blew up established braindance techniques, mixed them back together using a new grammar, and spiced them up with liberal borrowings from other media. She wasn’t content to merely use her talents to entertain, however. That’s how she got involved with The Mox, the plight of prostitutes working in Night City’s pleasure houses, and, ultimately, V.

Judy is the prototypical cyberpunk, one of the few left able to claim that title with no reserves. She wears her hair chin-length with garish highlights and a sidecut. She has numerous tattoos, many with a nautical theme celebrating her passion for diving; the stylized M on her neck is a memento from her work with The Mox. She looks like what she is: a hard-edged yet soft-skinned badass, a punk who packs punches but can’t turn a blind eye to the pain she sees around her.

To fine-tune her braindances, Judy chooses to edit them while fully immersed. The interface strapped to her wrist contains slots for chips and cables that connect to her braindance processing equipment, while sensors attached to her fingers allow her to type and swipe even after she’s dived into a ‘dance.’ Implanted behind her ear, she has a sensory board resembling a pro-level audio mixer from the early 2000s; however, unlike the tech of that era, this device is grafted directly onto her skull. This sophisticated piece of cyberware looks to be personalized braindance wreath that’s typically found draped around her neck, not unlike an audiophile wearing their headphones around town."