I wouldn't necessarily give up on the LA property. I forget which one, but there was a podcast that had an investor that lived in LA that was creative with his financing/units and ended up with close to 20 units. I feel in a city like LA or Seattle (where I am currently) you have to spend a little more time than in a city like Orlando (where my units are) purely because of the market. If you're really analyzing 3 deals a day and not just looking specific properties you'd like to live in, then you should run into something that cash flows (even though it might be a very small amount or breakeven). So to your first question, I think it's completely doable. Search on here for some locals in LA and ask them how they did it.
As for investing out of state... Is there a particular reason you picked those states other than proximity?
I lived in LA for 2 years and starting doing my research on my own, moved to Seattle for work and finally decided to invest in an area I knew over in Florida. If you're going to invest out of state, I personally don't think it is all about how close it is, but more about how well you know the area and the market. I knew the Central Florida area much better than anywhere else and my biggest problem was finding a decent management company. So if you do choose out of state, I'd say make sure you know the area well and find a good management company. Also something you can find here. Good luck!