Btw what credit cards do you have? I'm not sure how familiar you are with credit card churning, but since no one has mentioned it, you should apply for all of these airlines' respective credit cards to maximize earning potential. I'd at least 2 of them since the consensus is that you should aim for top tier on 2 of these programs (totally doable given the number of miles you'll fly in a year).

E.G.
CitiAAdvantage VISA and/or CitiAAdvantage AMEX
Chase United Mileage Plus Explorer
Delta Skymiles AMEX

You might want to consider the higher end versions of the Citi AA and United Mileage cards, the benefits might be of interest to you with the amount of flying you're going to do (for instance lounge access). These cards will help you accumulate miles even faster when you use them to purchase tickets from the respective airline (for instance 2x) which you then use on partner airlines for your personal trips. One thing though is to check your company policy and make sure you are credited the miles to your FF accounts (and retain them) since they are reimbursing you/paying for these flights. I know some policies don't allow you to keep them, but they vary from company to company.

Overall it sounds like you can't really go wrong, its just a matter of maximizing what you can get out of your work travel. Curious, what do you do that allows you to fly back and forth to LA? I love LA and would be thrilled to be able to go every weekend (well maybe not every single one).