1. How many times a week should I water.
2. When should i fertilize for first time.

You should water no more often than once per week NORMALLY. Your situation may not be normal. Watering really depends on Mother Nature, humidity, fog, shade, sun, clouds, wind, grass type, mowing height, soil condition, soil type, and probably something else. The idea is to allow the soil/sand surface to completely dry out before you water again. But furthermore, you should really wait for the grass to tell you when to water. St Augustine can be very unforgiving if it dries out. Watch for the FIRST sign of drying anywhere in the lawn and then water the entire lawn. Use your judgement based on shade, wind, etc.

Start by raising your mowing height all the way to the top. If you don't cut any grass at all for several weeks, that's fine. Eventually it will all be up around 4 inches high. Mulch mow when it gets up to 5 inches high. Tall grass takes much less water than short grass. I have a low water use demonstration program in my yard. I have not mowed my lawn since October. Parts of the lawn have survived quite nicely with no supplemental irrigation since then. This lawn is on the edge of the Texas desert so it is not exactly a bayou here. If we don't get a surprise storm today or tomorrow, then I am about to water the last hold out grass. This grass is in full shade all day long. Without mowing it is about 20 inches high and has great color. My only point is that tall grass needs less water and possibly much less water. Short grass needs water all the time.

Start your watering with with one inch and see how long the grass can go without wilting. As soon as you see it wilting, water immediately. If it has been less than a week since you watered last time, then water longer this time. Try to go a full week without watering. Measure your watering rate with tuna or cat food cans. Time how long it takes YOUR sprinkler system to fill the cans. That is about an inch. Mine takes 8 to 16 full hours depending on how I set it up. Other systems only take 15 minutes. You need to measure your own and not make any assumptions based on mine.

As to fertilizer: are you interested in an inexpensive organic approach or do you want something like the Scott's 12-step method? Back in 2002 I switched to organics and have written considerably about organic lawn care. Go to the Organic Gardening forum and find the FAQs. At the bottom of the list is the Organic Lawn Care FAQ. It is all over the Internet lawn forums. Seems to be a good place for most people to start out. You can start with organics at any time.

I'm concerned about the brown spots in the grass. Can you post a picture? Is it possible the sod did not make good contact with the soil/sand underneath? If it has knit into the soil, then you might have a disease going on. That needs to be treated if that is what it is. St Aug has a tough time if it gets diseased, but that's why you bought Floratam.

You should be nervous. The grass roots are not deep enough...yet. Watch it carefully. If the water is down deep, the roots will grow to find it. And so will the soil's beneficial fungi.

Organic you can put out any day of the year.

With chemicals you should avoid the heat.

How does your lawn look now with respect to the original issue? Do you have any brown areas or are they cleared up now? And have you changed your watering?

St Aug usually grows 1 inch per week. If you mowed it down from 2 inches to 1 inch high, then you could be removing 50%. If it was mowed back from 5 inches to 4 inches, then you are only removing 20%.