2Macs,
Two pages of posts and everyone has provided advice I agree with, but I like these types of threads so I'll try and contribute.
My initial research indicates I need a minimum of a 12 volt panel to charge one battery. Do I need 24 volts or larger for two batteries to keep them up and also use power from them at the same time for lighting, water pump and heating, etc?
-I think this has already been addressed well. Your battery system is 12V and unless you go outside the box it will be 12V regardless of the number of batteries in it. When people refer to "12V solar panels" they are really something like 17V. A component called a charge controller will make sure the batteries are receiving the appropriate charge voltage. Higher voltage panels require different and more expensive controller technology (Maximum Power Point Tracking, or MPPT). Solar panels provide current to your system that is available for whatever you want, with anything unused by immediate demand going into your batteries for charging. If an immediate demand is for more current than the panels can provide, your batteries simply make up for the difference. At night when there is no solar input happening, your batteries run everything. Many consider the solar component to be primarily used as a battery recharger, including me.
Can I attach these inverters to the batteries in such a way to generate AC to handle a 22 inch LED TV, charge a shaver, a toaster occasionally, or a 900 watt microwave occasionally?
-Yes, absolutely. We do the same. As already mentioned, the high draw items like toaster or microwave are not conducive to being run off a battery, nor are your inverters of sufficient capacity to operate those items. A TV or recharging small battery devices is well within your inverter capacities, MSW or PSW issues notwithstanding.
Can I even do this? If I can, what size solar panel(s) do I need?
-Can of worms... Without having real applicable usage data from an energy audit, it can't be answered. Generically, plan on needing about 1-2 watts of solar for every 1 amp-hour of battery capacity. With 2x12V batteries, I'm guessing you're referring to typical OEM dual purpose marine batteries, you've probably got about 150 Ah of capacity.