Los Angeles, eh? It's very tough to get a CHL, there. CHL's are dispensed by the Sheriff of the county. Los Angeles county is a big one, meaning that you'd have a long, long way to commute if you chose to live in a neighboring county, depending on distance. (Be mindful of the term "gridlock" and the fact it simply had to have been coined in southern California by someone viewing the morass of traffic during the rush hour times.)
Other counties are far simpler, including Orange County (just to the south), which would mean you could live in O.C. with a CHL and simply commute into Los Angeles with the other drones (via the freeway congestion).
As with any other state, if you get a CHL in the state then it applies across the whole state. California requires you to get approved by the Sheriff in the county where you reside. It's a might-issue state, subject to the predilections of the Sheriff, including whatever limitations he/she deems fit to add to your permit.
California also has a long, long list of firearms that are verboten.
There are also requirements regarding safe storage, and really punitive ones related to whether minor children gain access.
There's a maximum capacity (10rd magazines).
There are dozens of other little mines in the minefield, too.
As for importing guns you already own ... I don't know. Check with the Sheriff, or perhaps a competent attorney (on California firearms laws).
I would suggest contacting the Sheriff's department in Los Angeles, explaining your situation and asking about your questions.
The Attorney General's web site also has a decent summary (though, as of 2006's laws):
click. Read them and know them.
As well, get a copy of this book:
How To Own A Gun In California And Stay Out Of Jail, by John Machtinger. It'll help you spot the mines in the minefield you're about to enter. Keep in mind that fouling up on weapons charges can go on your record permanently ... and, worse, deny you your "right" to carry a concealed weapon permanently.
BTW, I carried in California for years, but in a rural county where it was practically shall-issue by virtue of the Sheriff's position regarding citizens being armed for self-defense.