Here are a few answers. I have to assume you either know a little bit about home electricity or are a fast learner.
1. To estimate battery run time, you need a few figures. First of all, you need an estimate of how much power you consume. This figure is NOT your peak usage, but rather your average usage. It can be hard to guess and can vary drastically from household to household. Perhaps the best way is to look at your electric bill. If you used, for instance, 1,000 KWH for the month and there were 30 days in the month, you divide 1,000 by 720 for a figure of 1.38. This would mean that in any given moment, you are using 1380 watts of power (1.38kw). Of course, this assumes that your meter gives a correct reading and that your inverter will be running everything in the house. Here is a battery life estimator from Xantrex (parent company of Trace) to help you figure how long your batteries will last.
http://www.xantrex.com/support/howlong.asp
If you are unable to use the above method, perhaps the next best would be to try to calculate the wattage of appliances in your home and try to guess how much usage they get per month. It can be tough to figure, but an educated guess is better than nothing.
To find the Amp/Hrs of your battery, you can go to
http://www.trojanbattery.com/ and either find the battery you are planning to use or find a similar battery. BTW, you will see that not all batteries are the same. Even though most batteries sold in the country from Trojan and Trace are similar, there are some differences and it can pay to pay attention to the technical specs.
After you have the battery run time, calculate enough batteries so that they won't be drawn down more than 50% on a normal disharge. How many more than that you want depends on you.
Also note that there are larger batteries than the typical golf cart batteries. They would allow you to use fewer batteries. You would just have to check availability and costs.
2. In order to know if you can run what you want to, you will just have to run the figures for your application. Also, when you install the dual inverters, you will have to try to load balance your circuit breakers so that each inverter has roughly the same load. Full size washing machines can run on some Trace inverters but they do take a lot of juice to spin including a pretty heavy initial startup.
If you are pushing the capacity of the 3.6, pay attention to the bypass circuit rating. When the inverter is off/charging, you electricity will still be passing through the inverters to get to your house. Even though the inverter is not running, it can still pop a fuse leaving you completely in the dark until you reset it. For the trace 3.6, I think it is 30 amps. In my opinion, it is undersized on that model. You can actually get more power from the inverter with it running than directly from the street with ithe power on. I have a Trace 3.6 and it has never shut down from overload on the inverter, but the bypass has popped once or twice.
3. I have never known the modified sine wave to be a problem for that type of load. There may be some electronic equipment that is sensitive to it and can even be damaged. Also, you should know that appliances that are normally silent will likely make a humming noise that may be quite subdued or quite loud. Ceiling fans, some floor fans, and microwaves are all likely candidates for a "hum." I have not known refrigerators, washers and other such loads to be affected by the harmonic distortion. As far as affecting your AC, it will probably run better off the inverter than the street power.
Also, please note that I am not recommending an inverter for your situation. I am only telling you that it can work! It might not be cost effective. Generally, for smaller backup load, inverters are cheaper than generators as well as being quieter, cleaner and generally more transparent. For larger loads, generators are generally considered to be more cost-effective. Also, a generator is easier to install and more portable than a huge inverter system but at the same time less likely to be stolen because of the sheer weight and size of the unit.
You will also have to consider the condition of the wiring to your house. If it is very bad, it might not support running your house and at the same time charging your batteries. With two inverters it should help since the charge should be distributed more evenly between the two phases. With a one large one phase system it is problematic since basically you have everything in the house on one phase plus you are trying to charge the inverter from that same power source.
In my house, I have a Trace 3.6 that runs everything but my 2 airconditioners and electric hot water heater including a large regrigerator, microwave, washing machine, deep freezer, computer, etc. We try not to use the iron, microwave and washer at the same time. The maid is trained to do the ironing when the power is on unless it is an emergency to cut down on inverter use. I would probably be happy with a 2.4 considering that it is a continuous usage rating and that the peak rating is far higher for startup loads. Also, the bypass circuit is 30 amps in the 2.4 as well as the 3.6. I have 16 batteries which I estimate would give me a runtime of 18-36 hours. They have only run out once since I installed them over a year ago and that was probably because I added water during one of the longest power outages of the year.
I would have bought 12 batteries, but I thought I might later put a small 110v airconditioner on it. As it ended up, I am glad I have 16 batteries, but I probably will never put an airconditioner on it unless it is one of the portables ones like PriceSmart is selling now.
I expect my batteries to last 3-5 years. They are over a year old and show no signs of weariness. I only have to add water about every three months since the batteries aren't worked that hard.
One last word, you pay a premium for electrcity from your inverter. It really does take more electricity to charge it than you get from it. Efficiencies vary by model. Trace inverters are among the most efficient. Also, it takes some electricity to maintain all of the batteries in a state of readiness (float).
BTW, there is nothing crazy about your idea. It is completely workeable, and there are valid reasons to do it. As you know, it isn't the usual route or necesarily the easiest or the most economical.