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 :-).
It sounds like you got the charging description of a lead-acid battery. Other battery types need to be charged in different ways.
For the lead-acid case that you describe, all you need is a single power supply that is both voltage and current limited. Initially the battery voltage will be low, so the power supply current limit kicks in. Eventually the voltage reaches the float level, so the power supply starts regulating voltage and the battery draws less current than the limit.
Best Answer
My best "educated guess" is that if you are discharging a battery, the "heating effect" is decreasing with time, whereas when you are charging the battery, the heating effect is increasing with time. So, as time passes, nothing happens to the battery when discharging. But in the second case, most likely the battery would explode, or at the very least, burn up! This still can happen, even at the lower charging rate if the battery can't dissipate the heat generated at this current.