You can seed immediately. You do need to get the dead stuff up though so the new seed comes in contact with the soil. If you have a bunch of little spots, you might spring for one of the all-in-one patch products. It is hard to keep a bunch of little areas watered well enough to get grass started.

I'm gonna disagree with dsa a tad. I'm in mid Ga and a healthy sa lawn up here never really goes dormant, stays about halfway green through the winter. I fertilized mine the beginning of april after I had already had to start mowing it and it being 100% greened up. I'm already on my 3rd or 4th mow. I imagine down in florida you had a head start on me. We had a very mild winter where i'm at. You are good to go, plug away. I plugged half my backyard last year on 18in centers, it filled in by fall easily. I put a tablespoon of starter fertilizer in each plug hole before setting the plug in. I will agree, stay away from weed/n/feed products, waste of money.

You have centipede in California? Is that a new trend since the drought? 99% of all the centipede is in Florida and still that's not very much centipede. It grows best in terrible soil with terrible care. It needs almost no fertilizer. People say that once every other year you can drag an empty fertilizer bag over it and that's enough. It loves the acidic soils like they have on the east coast. I would think St Augustine and bermuda would be the main go-to grasses in the Central Valley. Some water restrictions are fairly enlightened and some are guaranteed to kill everything if you follow them. Ours are horrible. We can water from 7am to 9am and 9pm to 11pm on one day per week. Well, who's around in the morning from 7-9??? You'd have to be retired. I suppose you could use timers (and I have a good one). But timers don't work for veggie gardens and plants located around the yard. That leaves 2 hours in the dark to get your watering done. In San Antonio we could water from midnight until 10am and from 8pm to midnight on your day. Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

I use Milorganite in the spring as well, for summer and fall I use the Scott's Turfbuilder product designed for that season. I am by no means a lawn expert though, these are just easy to find, relatively inexpensive products that do the job well enough for me. I'm confident you could kill the ivy (and not your trees) if you have the time and patience. Again I'd suggest trying the Weed B Gon Chickweed Clover & Oxalis concentrate in an Ortho Dial N' Spray hose end sprayer. Just pick a calm day and get down low to avoid spraying the tree trunks and leaves. If the ivy is anything like wild violet it will take a couple of applications but it will work eventually. Then like you mentioned you'll have to rake out the dead ivy and reseed the bare patches. But as others have said, a weed-free (or even weed sparse) lawn is a huge commitment so I don't blame you for hiring it out--if you can find someone to call you back. Last thing I'll say is that no matter how diligent you are about your own lawn, if your neighbors aren't equally diligent you will always have some of their dandelions, clover, ivy, etc. filtering into your lawn. So I've lowered my expectations and live with a semi-weed free lawn after 13+ years at my current house.