Many of the Radio controlled power points use the SC2262 / SC 2272 chips for the encoder and decoder.
The protocol is very simple and easy to reverse engineer.
You need to pull apart one of the power points to see what the decoder chip inside is.
Some clues that help are the number of power points that can be switched from the encoder.
If its 4 or less, its likley to be the above chips, but if the encoder supports more than 4 , then its a differant chip.
Ive had no success reverse engineering the power points where more than 4 are supported as the manufacturer rubs
the numbers off the decoder chip, and the protocol is very complex.

Whats the power consumption of the gyro and accelerometer.

Its certainly possible to do.
You would need to examine the HT-12E encoder spec to determine the protocol used
and then write your own decoding program.

ATX supply needs a minimum load on its output to regulate.
If you dont have a load, the output voltages will rise above their nominal values.
Depending on what the switcher chip in the supply is , the supply can shut itself down
if the output voltage goes high.
Depending on the power rating of the supply, you will have to experiment to find out what
minimum load is needed.

The Solar Idea is OK, but just use rechargeable NIMH batteries instead of Lithium Ions.
NIMH batteries can withstand overcharging without damage provided that the charge current is C/20 or less.
Using 6 AA 2500 mah batteries will provide between 7.2 and 8.4 V which is just enough for the 5V regulator on the Arduino board.
If you want plenty of battery reserve for cloudy days , then use C size nimh batteries.

What exactly are you trying to do?
Most LIPO battery chargers for more than one cell need some kind of balancing capability
to ensure that each cells voltage remains the same as the battery is charged.
This involves not only monitoring the individual cell voltage, but also controlling the current thru each cell.
Whilst this may sound easy to do its anything but easy, due to the multiple voltages and currents that have to be individually controlled.
As a result , there are dedicated LIPO battery balancing ICs, which do all the hard work for you.
Something like this
http://www.intersil.com/content/dam/Intersil/documents/fn82/fn8246.pdf

Texas Instruments and Maxim also make LIPO battery balancers.

Is the boost converter going to be permantly connected to the batteries?
The MC34063A has a minimum idle current of 4 ma so will flatten 2 AA batteries
in about 3 weeks , even with no load on the output.

What would you do with one.
Produces 69 picowatts of light.
Not much you can do with that, let alone see that it was on.

You need to determine what bluetooth profiles your phone supports.
Most phones with bluetooth support audio, file transfer and HID, none of which sounds very useful for what you need.

Yes, the module you have is a Serial Module.
Thats an unusual profile for a phone to support.
What exactly are you trying to do ?

What exactly is the PIC supposed to do.
You have set Port A and Port E to inputs.
Are you setting all the pins of Port A and Port E to some combination of logic levels.
The code seems to need a combination of logic to make something happen.
If you set pins to input, then you must connect either a 1 or a 0 to all the pins, or enable pullups ,otherwise they will float
and give erratic results.

OK, so whats the relative drain caused by the load , compared to the size of the Solar Panels.
ie how much is the DOD of the battery overnite.

Ive made a number of MPPT solar chargers, and you have the right idea with the various voltages and charge currents.
The practical problem that arises is that the input power to the charger varies over the course of the day and may vary
rapidly as clouds move over the solar panels so the tracker will be trying to maximise the power into the battery all the time.
Its quite possible that there simply wont be enough power available to provide the bulk charging and equalizing currents
when you want them, so you have to keep track of how much power was available to provide the required charging when you wanted it.
Is the battery that is being charged also being drained by some load, or is the Solar Charger just to keep a battery fully charged by trickle charging it.

Id test the 2 output devices, they look like transistors, with a multimeter for collector - emitter for short circuit
and if good , replace all the dead electrolytic capacitors, and turn it on with no load.
If it works, a cheap fix, if not then its likley that the fault will be hard to find , unless you are really keen.

Unless the PS is a switchmode type , and most arnt , it will simply be a transformer, rectifier bridge, and
some kind of linear regulator, using either transistors or Fets, most likley transistors.
Easy to fix.
The only expensive bit is the transformer , and that wont be damaged by incorrect volts on the output.