Monday 4 pm from Beverly Wilshire to Staples Center will be about 1 hour via Olympic Blvd. You can try I-10 but it will take about 25 minutes just to get to I-10 from Beverly Wilshire. Not worth the trouble in my opinion. Take the local road (Olympic is the best).
Staples to LAX at 9 pm mid week should be about 35 minutes via I-110 and I-105. That's your best route.
Go Kings Go!