Any reason in particular?
What do you like about vine and cahuenga?
Also you mentioned looking at koreatown, would you consider that a good alternative?
Strange that you say not to bike anywhere, while I was there it felt almost like Europe with dedicated green bike lanes in some areas. But maybe it was just those few areas
Burbank Airport (and Long Beach) are just way easier and less congestion. The entire loop system at LAX is fucking broken and awful. People everywhere. Constant back up jams. Lines everywhere. Most terminals have pretty mediocre food options, and the entire place is under construction. Burbank and Long Beach are heaven compared.
Vine and Cahuenga area are rapidly gentrifying (both good and bad). There's a lot of great bars and restaurants within walking distance of one another and it's a very young vibe. It's super hipster on Cahuenga, like a mini one street Portland. The music scene is big around the area with the Palladium, Fonda Theater, and various small night clubs scattered off Hollywood Blvd, so that's always entertaining. Plus Amoeba Music and lots of movie theaters (Art Light, El Capitan, Chinese Theater). But there's homeless, trash, and foreigners intermixed throughout so be weary of that. Plus things are semi-over priced due to it being a tourist hub.
As for Koreatown I don't know as much about it. Certain parts are REALLY run down but there's a lot of diversity there, in terms of food, culture, people, and things to do. Koreatown's proximity to Downtown is a big plus, but the drawback is there's no solid train lines in the heart of Koreatown (Purple line borders the North side). There's a lot of interesting bars, clubs, restaurants, and weird shit to do, but again I just haven't tried much of it. Maybe someone else in this thread can speak to it.
You mentioned for your bike you were in Santa Monica, that likely gave you a skewed view of bike friendliness of Los Angeles. Westwood/Brentwood/Santa Monica are slightly more biker friendly because there's a lot of money running through them so they have expanded streets to include bike lanes and have developed more infrastructure. But again based off your expense requirements you can't afford to live out there. Most places in LA are a fend for yourself mentality on the road and literally no one has patience for bikers. Some might argue it's doable, but I moved from Portland which is the bike capital of the United States, and it's like night and day difference. I would die out here on a bike, guarenteed.