Advice - Budget - Neighborhood

Hello,

I will be moving to LA for work from another state in late August. When do you suggest I start looking for an apt?

I will work only two days from the office and the rest from home, so I am looking for a two-bedroom apartment so I can make the spare room as an office. It must be cat friendly!

I am looking to spend $3k maximum but would prefer to stay at $2700-$2800 ish. Is it reasonable for LA?

Also, I will work in downtown LA, so I am looking to live not too far from it. It would be just a two-day commute per week, but I keep hearing how insane the traffic in LA is, so I am concerned. Which good neighborhoods do you suggest?

Thank you!