Hi,
1) I think that you probably have to allow at least 10x surge over your fridge rating for the inverter not to trip. And that's thick battery wiring too.
2) If you have something that is interconnected with the mains, and worse, has the possibility of back-feeding it if you make an error, then you will likely need at least a part-P-certified electrician on the case, to be legal and to avoid (say) invalidating you house insurance if it happens to burn down once you've set your stuff up.
3) I have 5kWp+ on my roof, fully grid-tied. I have about 200Wp off-grid which mainly manages to power my super-efficient *4W* SheevaPlug Internet server. Daily insolation in the UK averages about 2.5Wh per Wp of panel, down to 1 in winter, up to 5 in summer; less if not pointing due south and with optimal tilt. Work out the actual consumption in Wh/day to work out what fraction of time they would be solar powered. Appliances can be astonishingly energy-hungry unless designed otherwise. A minimal cup of tea takes about 30Wh to boil, as a data point for you.
4) Voltage without a load is, I think, an unreliable measure of available power from solar panels.
5) I have from time to time used an off-the-shelf (CE-marked) power adaptor that will accept power either from mains or 12V (eg designed to work in car), and switched the 12V or 230V side (eg with SSR) when the battery voltage has been high enough, eg see here:
http://www.earth.org.uk/low-power-laptop.htmlRgds
Damon