Edited 2017 - changed recommended long life storage voltage and added comments on fast charging using some recent systems. RM.
What YOU do as regards several of these questions depends largely on what YOU are trying to achieve or test.
Discharge to cutoff is fully discharged (to whatever remaining % that voltage represents). That's the easy one :-)
Percent dropoff of current in tail sets final % of max possible charged reached. There was a superb table given here within last week or so. Can supply later if you don't find it.
Real Men™ plateau at 4.2V and tail down to 10% or even 5% of the constant current rate. This gets the battery full and knocks the stuffing out of it.
Others terminate the current tail at say 25% of cc value.
Optimum lifetime for ongoing usage is at about the end of the constant current phase. That makes it very easy to locate - charge at specified current until desired max voltage is reached, then charge at constant voltage as desired. Here "desired" is to stop immediately. This is the point at which batteries tend to give significantly longer whole of life mAh of storage without grossly reducing mAh capacity per cycle. This is liable to be the point where older "fast chargers" tell you they have finished. Actual % total claimed varies but probably 70% - 80% range.
Newer USB input fast chargers use the term differently. In the case of USB the maximum available charge current at 5V is 5A so that the battery MAY be able to be charged at ~= 6A for the CC part of the cycle using an efficient buck converter to drop voltage and raise current.
[For a buck converter: Vout x Iout = Vin x Iin x efficiency_of_conversion]
Some systems such as QuaqlComms Quick Charge system allow the use of higher charger voltages (9, 12, 20) with specifically designed equipment, so battery charging can be faster for a given voltage provided that the battery specification allows this.
Maximum charge rates for LiIon and LiPo batteries are usually C/1 = 1A per Ah of battery capacity.
At 5V, 5A a USB charger can charge a 6000 mAh 1 cell LiPO battery at max rate - so eg a 10,000 mAh single cell battery used in some larger tablets can not be charged at the allowed 10A ! rate.
For long life storage where actual stored capacity is unimportant, LiIon and LiPo cells should be stored at about 3.7V.
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Using cells without protection adds to the rich tapestry of life. As long as you don't mind the occasional scorch mark on the tapestry that's fine. Note that part of the protection is a one time high capacity fuse under the cap for when things get out of control. Undervoltage discharge destroys. Charging from below a certain voltage at full rate can get fun, I'm told. Charging at reduced rate can bring cell up, I'm told. Below another second level they say don't even think about it. I've had very poor success in trying to get LiIon to misbehave. I have a box of unprotected cells that are very uncooperative about venting with lame etc. Strange. Sony and Apple and even HP seem to be much better at it :-).
Batteries are tricky nonlinear devices. If you want to build a safe and reliable lithium battery charger, you need to know more about your batteries and your battery charger circuit. In general, lithium batteries are not interchangeable and not all chargers will work with all batteries.
There are also significant differences between lithium (probably not what you mean), lithium ion (probably what you mean, sometimes written Li+ or Li-ion), and lithium polymer (sometimes written LiPo) batteries, and significant differences within the battery chemistries of these categories.
The datasheet I found for TP4056 does not say that this device includes undervoltage protection. It only provides charging. Perhaps other circuitry does on that evaluation board does. The appropriate discharge protection is a function of battery chemistry, and the threshold should set according to the battery manufacturer's datasheet (as a baseline, anyway).
A circuit that tests for undervoltage is probably measuring the battery under load, and will need to compensate for the battery's internal resistance \$R_{internal}\$ reducing the voltage at the terminals \$V_t\$. That is
$$ V_t = V_{oc} - IR_{internal} $$
So to answer your second question it's quite possible that \$V_t\$ = 2.5V is an appropriate cutoff for a battery with \$V_{oc(min)}\$ = 3.0V, if \$ IR_{internal} \$ ~ 0.5V.
Best Answer
The available capacity of a fully charged battery varies with temperature and discharge rate (and duty cycle if it's not constant). It will tend to decrease with each cycle, and previous fast charging, exposure to temperature extremes or fast discharging may have negative effects over time.
If a reputable manufacturer says it is rated at 20Ah that means that under the specified conditions you can get approximately 20Ah from a new battery before it discharges to the point where further discharge may cause damage. The details of how it behaves should be described in the datasheet and application literature. You may wish, for example, to charge slower than the maximum or discharge only to a higher voltage than the minimum in order to extend the battery life. The latter will, of course, reduce the available capacity.
If you'll forgive an analogy, your question is a bit like asking about a car manufacturer that claims 7 liters per 100km (33.6 miles per US gallon) fuel efficiency and whether that means you'll actually get that efficiency. It depends on a lot of factors, such as your driving habits and the condition of the automobile.