1. Find an empowering female video game character that's somewhat pretty, yet not unrealistically and overpoweringly sexy, with no negative implications of femininity or clichés.
Chie Satonaka (Persona 4.)
Disclaimer: I actually don't like Chie as much as most people. I'm one of those vocal anti-Chies who had absurd accuracy issues where she'd only hit 1-2 out of 5 enemies with a multitarget attacks, and I also found her social link to be not that interesting. But I kinda like her in a way. Just not in the capacity as her fanboys.
You might argue that she fights in a skirt which is sexist, but the thing is, it's her school uniform. I was surprised to learn this back in high school, but many schools anally enforce "you MUST wear your uniform until you get home" and give detentions or worse to offenders, so the cast wearing their uniforms into battle isn't quite as stupid when you consider that... And even then, she has shorts underneath, so it's not that bad. If anything, as a fan of kung-fu who trains a lot and favors her feet in combat, this skin between the socks and the shorts actually shows off well toned legs, which makes her more appealing without being piggishly sexist.
The game makes a point about her being pretty. She has her own fan club of sorts, despite Yukiko's being bigger. But the reason she fits this category is her personality, by far. She's always been standing up for various reasons, such as "The Vanguard of the Lunch Revolution" (which probably just meant she wanted steak on the menu, but hey, protein is kind of a must for a martial artist!) But more than that, she stares a small gang down and declines resisting since she fears they'll hurt her best friend. She's got balls. And not just the ones on her chest.
Don't get me wrong. I still don't ship her with the protagonist or ever use her in battle anymore (even before a full party, leaving her behind means more exp for the rest,) but despite her relatively bland social link compared to some others, if you look at the cast based solely on their merits in the main story, she's not bad at all.
2. Find a sexualized male video game character that caters to both the male and female sexual fantasy.
This is a harder one for all the wrong reasons. Thing is, I'm comfortable in my heterosexuality and I can comfortably judge whether a man is handsome or not - by a woman's standards. Largely since I spend so much time around women. But I can't really get into the mindset of a gay man to judge a character.
I'm taking a glance around at my game library, and honestly, I can't find any good examples. This is largely due to a looong period of unemployment and my game collection literally being 1/3 of what it used to be. But the thing is, most of my games are now family friendly since I babysit (what happens if I'm playing something like Duke Nukem Forever when my Nephew wakes up with a nightmare and comes to get me? Oh crap, right?) As such, being left with games like Atelier Totori/Meruru, Rune Factory, etc, doesn't leave me much in the way of actual "sexualized" characters, just male characters that my female friends find very hot, like Rex in Agarest 1 (admittedly not one of the more child friendly ones.)