This circuit uses a bi-phase rectifier. Each diode acts as a single phase rectifier but the AC voltage applied to D2 is 180o out of phase, thus full wave rectification is achieved.
Measure the battery voltage.
Your drawing doesn't show how the batteries are connected or the solar cell voltage. The battery voltage could be 12V, 24V, 36V or 60V, depending on how they are connected.
I don't understand the question.
Generally, you don't worry about the inductance of the secondary coils. Wind the primary for the desired inductance, then wind the secondaries based on the turns ratio, rather than the inductance.
There's no resistor between the output and adj pin. Normally 120R is recommended but as it's possible you'll be able to use a higher value as the fan will provide the required minimum load.
Although the oscillator will normally work with two gates, it may fail to oscillate if C is small.
A three gate oscillator is more reliable.
http://www.fairchildsemi.com/an/AN/AN-118.pdf
It requies a fair amount of processing power so you need a computer, not just a little microcontroller.
A Raspberry Pi with the appropriate image recognition software (there are probably some free/open source programs for this) is probably the cheapest/easiest ay to do this.
EL wire typically requires about 90VAC to 150VAC at 200Hz to 1kHz to light. A bug zapper produces over 5kV.
Your best bet is to buy an inverter specifically designed to power it.
Wow what a complicated way of doing it. Rather than a CA3130, two 555s and a speaker, I'd opt for an LM393 and a piezo buzzer.
Have you tried testing the astable part of the circuit separately?
Lol what amused me is he wasted 5 regulators without realising there was a problem with the circuit.
The lesson is, if it gets hot and components smoke, check there's nothing wrong with the circuit before replacing them and trying again.
Here's how to do it correctly.
It seems like an interesting concept.
The PDF you've posted contradicts itself. The specification says it's capable of driving 24 segments, yet the example in the photograph shows a display with 30 segments.
Have you measured the mains voltage?
Does it have an active power factor correction module? Some PSUs have a boost converter for power factor correction, before the main filter capacitor.