The in-battery protection circuitry is usually intended to act as a gross fault protector and it is strongly recommended that it not be relied on as a means of charging control. As a means of gross short circuit protection it may be suitable as long as the values they choose for max Iout are acceptable to you.
For charging, use of one of the large number of LiIon charger ICs is recommended.
A major factor is that the over voltage circuit does not remove the applied voltage when the CC charging current falls to a low value. This means that the battery is "floated" indefinitely with the risk (I'm told) of plating out metallic Lithium.
A PV panel (solar panel) that is nominally 12V rated and intended for charging lead acid batteries, will have a loaded Vout of about 18V and an O/C or light load Vout of over 20V.
The maximum voltage that you need AT the battery pack is 4.2V/cell or 12.6V in your case.
PV panel available Iout values are a reasonable approximation to being linearly related to isolation (sunlight level).
However, Vout is not related to light level in the same way. A PV panel will produce over 90% of its full power voltage for light levels of a few % of maximum and above - say at 10%+ to be safe. If you want the panel to charge the battery to fully-charged even on a low sun day, if necessary, then you need a panel that is full load rated at at least 12.6V/90% = >= 14V. As above, as an SLA targeted 12V panel makes about 18V at full-sun full-load, such a panel will provide more than enough voltage under all practical light conditions.
You will get substantially longer cycle life from a LiIon cell if you terminate discharge at a slightly higher voltage than allowable absolute maximum. With LiIon , below about 3V under medium loads you have used the large majority of the stored energy.In-battery low voltage cutoff circuitry will probably allow discharge to about 2.6V/cell, which is lower than is wise for good battery lifetime.
Well, This is an interesting problem that can be tackled a variety of ways. Typically to accurately measure the remaining capacity of a battery you will use a coulomb counter, which basically integrates charge flowing across a shunt resistor in both directions. Essentially this gives you current/time bidirectionally and you can solve for remaining capacity based on your initial specifications for the battery in your system.
For less accurate gauging (which is what I am assuming you are doing) using the battery's discharge curve to relate the voltage of the battery with capacity remaining can be done with a resistor ladder and comparators at each branch, like was previously suggested. This is from google images, so you will have to change the resistors to produce voltages at the junctions that correspond to points along the discharge curve that you will use, this is also a liquid level indicator, but the principle is the same This however will cause some problems if the battery is a rechargeable type since aging effects will reduce accuracy of the gauge as the battery is repeatedly cycled. The voltage/capacity relationship also changes as the battery is put under different loads, so either you can do the math to take that in to account or just measure the battery voltage when you are under a very light or no load, take a look at the picture for a typical AA battery discharge curve.
If you don't always need to check the battery, you can put a tactile switch in series with the circuit to kill power, that way you will only ever have the LEDS enabled when you are actually checking the capacity
Does the phone NEED a Lion battery? Could you remove it, and just throw in a USB cable with instructions on how to connect 5v to it? In 50 years 5v will still be 5v.
I'd worry what a battery (ANY battery) will do when stored/buried for 20+ years, it could leak and ruin the contents of the capsule, self-combust, leak acid that will eat through environmental protection of the capsule, etc.
As others have said very long life batteries are available for a price. You could build one of those into a basic USB charger so future generations can power the thing up, and toss the Lion battery.
Oh and remember to write down the unlock code :p