Strange that Hero Labs didn't catch the 1/2 Skill for Improved Con. I'll correct that.
The minor cosmetic biomods, according to Hero Labs, cost no essence or money, so I didn't list any. Because she's a dryad, I figured natural beauty would be acceptable for the day job I provided her. Is the essence/nuyen cost actually zero, or no? I have the credit chit that I can convert back in BP/chargen money, so paying isn't an issue at the present. Worse comes to worst, I can reduce some of the contacts to generate more money.
So I should up my Intuition and get perception. Not the easiest thing in the world to do... should I reduce my Agility to accomplish that, or switch Logic and Intuition? I can reduce Intimidation to buy Perception, but from what I'd read it's important to have a skill at 6 -- and Intimidation, since it isn't affected by glamour, seemed like a wiser choice. What's your opinion on the matter?
I'd like to think I have a good background, although I'm biased since, y'know, I'm the one who made it.

But to answer the questions in the order you've asked them in: functioning as a shadowrunner will probably be the hardest part, on account of her visibility. But that can also be a bonus -- she can get into places that normally wouldn't be open, she can provide a distraction with the equivalent of a tweet, and when she deigns to speak to someone, she can charm them to hell and back. You provided a build for a more subtle 'social chameleon' in a thread, and I can definitely see the line of reasoning that would say a chameleon would be useful -- but this is my first time in Shadowrun, and I decided I'd go for the flashy starlet. My hope is that she can leverage her fame and charisma for the betterment of the shadowrunning team. I intend to investigate options for false SINs or ways to obfuscate her fame in-game, which dovetails with her mental state (the effects of living a glamoured life -- it provides a unique world view, one I'm excited to explore).
As per her background, Lizzie's husband was a shadowrunner back in the day, before stealing the heart of the future Mrs. Xiao. However, he disappeared -- and, with Elizabeth's family's connections, she suspects foul play. So she spent a good amount of nuyen on private investigators, proper channels, all that stuff, but when nothing was turned up she determined her father was getting his revenge on her marrying 'down.' Elizabeth doesn't perform these black budget actions because she wants to get paid -- she's well off, her son will be well off, she's not worried about that. She just wants to find out what happened to the love of her life. All the luxury in the world means nothing when there's a gaping hole in the tapestry of your life, after all, and as a dryad she feels the effects of her husband's disappearance even more profoundly due to the effect it's having on her son.
I'm not sure who the other runners are yet, so I'm not sure how she'll interact. I imagine she'll treat them with a sort of perverse curiosity, seeing them as these mythical subjects of her childhood stories and the scripts she performs in her day job. There'll be a desire for connection, both because of who she is and because of who her husband was/is -- since he was a shadowrunner and refused to talk about that portion of his life, Lizzie is curious to learn more about that entire chapter of his life. Anything to bolster the connection she longs to feel again. But it'll definitely depend on the specifics of the other player characters.
It's supposed to be a "lighter" game. From the DM herself: "A few more reasonable authority figures, a chance to actually change the world. Less torture porn. Basically there's actually hope that things will get better. [...] As for not having a good time at a brothel? No those exist still, but they're not run by people who make the hunger games look like a comedy." Not sure what the other players are looking to play, though. The game is very much in its infancy. Sorry that it's a lame answer!
I definitely went into making the Eastern Rose with the understanding that she's not a typical Shadowrun character. But, in my newbiness, I'm hoping it's not insurmountable. It's all about the different avenues of getting Datachit A from Vault B in Company C, and how you go about it is the meat of the story. Hopefully having a socialite par excellence [at least, to the best of my abilities to make her!] will open up additional avenues, while the other 'runners will provide avenues of their own. The result, my hope is, will be a group that won't have the same standard answer to each question, but instead will have a whole basket of options to select a solution from.