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.

A circuit diagram would help.
The 16F887 code seems to be some kind of combinatorial logic that needs combinations of input pins to do anything.
Are all the pins that are defined as inputs connected to something?

Does the 16f887 have some kind of program in it ?

One trick you can use to minimise receiver consumption is to pulse the power to the receiver
on and off , where for example the receiver is on for 1 second in every 10.
This cuts the receivers power by 90% but requires that the transmitter has to transmit for
at least 10 seconds to ensure that the receiver in on during the transmission.
Otherwise its fairly hard as receivers have to draw power to operate, and most of the power is
the oscillators which usually run at high frequencies.

The battery wont come close.
The batteries spec is 610 mAH at 20C max discharge rate, which is 12.2 amps max.
The motors are specced at 7.5 amps each at full power which = max discharge rate of 30 amps for 4 motors.
The battery will be destroyed.
You need a far bigger battery, at least 1500 mAH @ 20C, or much smaller motors.

Need information about the Solar panel you want to use.
What voltage is it , and what is its power rating.

This chip MC34063 will do what you want.
Very easy to build and this tool http://www.nomad.ee/micros/mc34063a/
will give you all the component values.

This may help.
The voltage specs for these Transmitters and Receivers are a bit confusing.
The Transmitter will work with any voltage between 3V and 12V , but the Receiver needs 5V, and it needs to be
very close to 5V, otherwise the receiver wont decode properly, and this can show up as poor sensitivity / poor range.
Running the receiver off anything much above 5 V will damage it.

Its possible to buy commercially made watches that do all that the OP needs.
Mainly made for athletes for running or biking.
Garmin is the most popular.
Very hard to make something like this yourself and keep it small and lightweight.
Even with the commercial offerings, the GPS is what uses all the power and battery life is around 10 hours
with the GPS function on , much longer with it off.
The watches use LIPO rechargeable batteries.

The single biggest issue with battery projects is how long do you want the batteries to last
before they have to be replaced.
That effectively defines what sort of batteries are needed, and hence what voltage batteries are needed.
Id stay away from any kind of arranagement that needs voltage converters of any kind, because although they are more
efficent that linear regulators, at very low power levels , they can consume more power than the micro.

The problem with this idea is getting power for the Arduino.
A light switch is fed with 2 wires which go in series with the light being switched.
When the switch is off, then there is 240 V across the switch which you can power something with.
When the light is switched on by a relay or whatever, there is no voltage avalaible to power whatever is doing the switching.
You have to include a battery somewhere, plus a method of charging it .

No, running any kind of Arduino off AA batteries is generally a bad idea unless you
can take some kind of drastic power reduction type strategies such as sleeping the Micro,
or you are happy to frequently replace the batteries.
This reduces the power consumption whilst the Micro sleeps, but it also cant do anything
whilst it sleeps, apart from wakeup on specific conditions.
But this doesnt help if you are using a GSM module which draws power all the time.
Basically, you need to figure out how often the GSM module will be transmitting, as thats
the big power consumer, and that will determine the battery life.

Before you design a charging circuit for your robot, you need to know the robots power consumption.
If its well in excess of what the Solar cells can make , then its questionable whether the whole idea makes any sense.