What is the optimal voltage for storing a 3.7 V lithium polymer battery?
The $\pu{3.6-3.8 V}$ range is a good general choice, but it may be battery-specific. The particular voltage for 40% charge may differ for different cell technologies, e.g. various deviations of electrode materials and due to cell aging.
The optimal storage conditions, as you mentioned, are more often expressed as charge/capacity % ratio. Usually, the optimal storage conditions are said to have about 40% of the maximal charge. For longer term storage, it is advised to be kept in a fridge, but not to get below
$\pu{0 ^{\circ}C}$. Note that by the semi-empirical van 't Hoff's rule(*) chemical reactions (= battery deterioration) speed up typically 2-4 times per temperature increase by $\pu{10 ^{\circ}C}$.
Lithium-ion/lithium polymer (Li-ion/LiPo) cells reportedly lose typically 20% of capacity per year when fully charged, but just 4% at 40% charged state.
For practical purposes, sometimes a trade off is made between battery lifetime and battery readiness.

Some background about Li-Ion/LiPo cell chemistry:
The negative anode is formed by lithium graphite intercalate $\ce{Li_\mathrm{x}C6}$, x&lt;=1
The positive cathode is formed most commonly by $\ce{Li_xCoO2}$
The electrolyte is most commonly formed by $\ce{LiPF6}$ dissolved in mixture of dialkylcarbonates $\ce{R1-O-CO-O-R2}$ (**)
About electrode reactions from Li-Ion wikipedia article:

The positive electrode (cathode) half-reaction in the lithium-doped cobalt oxide substrate is
$$\ce {CoO2 + Li+ + e- &lt;=&gt; LiCoO2}$$
The negative electrode (anode) half-reaction for the graphite is
$$\ce{LiC6 &lt;=&gt; C6 + Li+ + e^-}$$
The full reaction (left to right: discharging, right to left: charging) being
$$\ce {LiC6 + CoO2 &lt;=&gt; C6 + LiCoO2}$$
The overall reaction has its limits. Overdischarging supersaturates lithium cobalt oxide, leading to the production of lithium oxide, possibly by the following irreversible reaction:
$$\ce {Li+ + e^- + LiCoO2 -&gt; Li2O + CoO}$$
Overcharging up to 5.2 volts leads to the synthesis of cobalt(IV) oxide, as evidenced by x-ray diffraction:
$$\ce {LiCoO2 -&gt; Li+ + CoO2 + e^-}$$
In a lithium-ion battery, the lithium ions are transported to and from the positive or negative electrodes by oxidizing the transition metal, cobalt ($\ce{Co}$), in $\ce{Li_{1-x}CoO2}$ from $\ce{Co^{III}}$ to $\ce{Co^{IV}}$ during charge, and reducing from from $\ce{Co^{IV}}$ to $\ce{Co^{III}}$  during discharge. The cobalt electrode reaction is only reversible for $x &lt; 0.5$ (x in mole units), limiting the depth of discharge allowable. This chemistry was used in the Li-ion cells developed by Sony in 1990.

In addition to the above, there is another deteriorating process, related to keeping Li-Ion/LiPo cell long time at fully charged state ( typical for laptops permanently connected to AC, unless special coutermeasures are involved ). There is a side tendency of lithium ions in the highly charged cell state to form metallic lithium at surface of graphite-lithium intercalate. Normally, this is reversible. But there is ongoing slow side reaction with the dialkyl carbonate solvent:

The Reaction of Lithium with Dimethyl Carbonate and Diethyl Carbonate in Ultrahigh Vacuum Studied by X-ray Photoemission Spectroscopy
Abstract: The reaction of dimethyl carbonate (DMC) and diethyl carbonate (DEC) with clean metallic lithium in ultrahigh vacuum was studied by the use of X-ray photoelectron spectroscopy with the temperature-programmed reaction methodology. Both molecules are of interest as solvents in ambient-temperature lithium batteries. The solvent molecules were condensed onto the surface of an evaporated lithium film at $\pu{120 K}$, and spectra were collected as the sample was warmed in ca. $\pu{25 K}$ to $\pu{30 K}$ increments. The reaction of either DMC or DEC with lithium was initiated at $\pu{180 K}$, a temperature much lower than their bulk melting temperatures, producing lithium methyl carbonate, methyllithium and lithium ethyl carbonate, and ethyllithium, respectively. At temperatures greater than $\ce{270−300 K}$, the lithium alkyl carbonates start to decompose with $\ce{Li2O}$, elemental carbon, and alkyllithium as products on the surface. Both DMC and DEC are more reactive toward metallic Li than another carbonate solvent, propylene carbonate, which we have studied by the same methodology. Because methyl and ethyllithium are highly soluble in the parent solvent, electrodeposited $\ce{Li}$ is predicted to have poor stability in an electrolyte composed of either DMC or DEC.


(*) Not to be confused with the van't Hoff's equation - the van't Hoff's rule is about temperature dependency of kinetic rate, the van't Hoff's equation is about temperature dependency of equilibrium constant. But both are related.
(**)( dimethyl carbonate, diethyl carbonate, methylethyl carbonate, ethylen carbonate, propylene carbonate )
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Being fully charged helps to prevent deep discharges, what is not the same as saying it is optimal storage condition. Note that storing cells fully discharge is worse than fully charged.
About half charge is optimal for lifetime and full charge is optimal for the time before needed recharging. But self discharging is not dramatic. LiPo have reportedly self-discharge rate approximately 5% per month.
This discharging is probably not linear, and its rate would be decreasing. Also, consider it as illustrative number only. It may differ for different cell technologies and age. Discharge cutoff cuts the connection before a real 0% charge. So the effective 0% charge &gt; real 0% charge. Generally self discharging rate may increase with battery age, but it can be an individual number.
The usual mode of Li-ion/LiPo charging is by a constant current, until the threshold voltage is reached. Since then, charging switches to the constant voltage charging, until the charging current drops below a threshold value. The point of mode switching is typically between 70 (old cell)-80(new cell)% of the full actual capacity.
All that means the charging current is constant until the charge mode switching, and then it is asymptotically decreasing toward zero, until the current threshold cutoff stops the charging.
For lifetime, more shallow charging cycles is better than fewer deep charging cycles. The former has bigger total provided energy value for the cell lifetime. E.g., just illustratively, if a cell lifetime is 300 0-100% cycles, it has lifetime e.g. 800 30-80% cycles, giving 4/3 times bigger total charge. And it goes further for even more shallow cycles. But, unfortunately, shallow charging strategy is seldom practical.
So, if you know you are going to use the battery shortly after charging, it is the best to fully charge it, or at least to 90%. If you do know it will not be used for some time, it is better to charge the about discharged battery just partially, e.g. just 30-40% of the typical full charging time, depending on the residual charge.
The charging time depends on the battery state. A new one may have bigger capacity and generally different remaining charge. Note that new batteries are shipped partially charged, never empty. So bigger free capacity = longer charging time. The same free capacity, but older battery = longer charging time, because the second stage kicks in sooner due higher internal battery resistance.
The discharging cut-off voltage is vendor and technology dependent. It is usually somewhere between 3.0-3.5 V. Note that voltage as a function of charge is strongly nonlinear. More discharged = steeper voltage fall. The same vendor and technology dependency is for the cutoff charging voltage. The percentage itself is hard to tell, but estimated from the voltage curve just few percents of the capacity, perhaps 2-3%.