Hi OP, I've been living in LA for the last 2.5yrs with my 2DC and DH.
Schools wise yes everywhere is catchment, but there are ways of applying to schools if you love outside the catchment. However it is stressful and over subscribed so I'd highly recommend doing schools research on greatschools.org then renting something in a good schools boundaries.
Definately rent close your DH work - the work ethic for most industries is hard so you don't want to be stuck at home most evenings on your own because it takes him 2 hours on the freeway to get home!
Los Feliz and Silverlake are great neighbourhoods but like everywhere in LA they are neighbourhoods. Most people live in apartment blocks or bungalows (single family homes) so everywhere outside of Downtown feels surburban. It took me a long time to get used to that! I live on the Westside so don't know Burbank that well but it is pretty surburban and dull. NoHo less so.
I have a few friends who homeschool and are members of various homeschooling groups so if you go in that direction I'd be happy to put you in touch with them.
LA is a much better city to live in than visit, there's so much to do and you can get out to deserts and mountains and Unspoilt beaches (and Mexico!) very easily.
Be wary of BritExpats....they are in general pretty cliquey and negative to newbies so just take everything with a pinch of salt! Glean the facts and move on....
Feel free to PM me, my DC are younger than yours but my eldest is about to start kindergarten so I've just been through the whole schooling process and as I say I do really enjoy living in LA - once you get over the unexpected culture shock!