Are there important points to consider in typical or special applications when capacitors operate with applied voltage close to their rated DC voltage? Such as:
Will the capacitor's life time get shorter, and has this been verified somehow, or is it only a myth?
It depends. Capacitors are fairly complex components, despite their appearances, and quite a bit of consideration goes into applying them such that they last a long time. The below covers only a few points, and I'm not even touching the foil capacitors.
Tantalum capacitors must be used well within their rated voltage, pulse/ripple current and temperature ratings. They don't usually fail gracefully, but short out. They will usually hold their capacitance within tolerance until the very end. If the circuit is properly protected and the capacitor's failure resistance is low enough, nothing bad happens other than the circuit not working anymore. But often the capacitor will fail not as a "dead short", but with enough resistance to heat it up and ignite it. They burn for a short time but at a very high temperature. It's a catastrophic failure.
Electrolytic capacitors fail "slowly" and typically, but not always, tend to lose capacitance and become more and more resistive. Eventually, they end up as low capacitance capacitors with very high ESR - the approximate end condition is an open circuit. An aging electrolytic may also develop low impedance between the electrodes - not usually a short, but a resistance in the right range to turn the capacitor into a heater. The self-heating boils the electrolyte and causes it to vent and corrode anything it comes in contact with.
The electrolytics that fail low-impedance may appear fine with no DC bias, i.e. if you check them out on an LCR meter. The failure may be triggered only after the capacitor voltage has been brought up high enough. I've had a few large electrolytics from lab power supplies that behaved that way: mostly nominal at low voltage, shorts out once some threshold voltage is passed.
The biggest killer of electrolytic capacitors is heat - whether due to ambient temperature, nearby heat sources such as hot parts and heatsinks, or self-heating due to the ripple current. Remember that the electrolytic capacitor life ratings are at the maximum rated operating temperature. An electrolytic capacitor that's running cool - with internal temperature of, say, 35C - will retain usable ratings for tens of thousands of hours if it was well made. They are more likely to fail shorted when operating near the rated voltage and hot, but at least in my limited experience that was not a problem when running them close to room temperature. Using a 16V capacitor on a 15V circuit may be acceptable if the capacitor is cool during operation and the 15V is well regulated. If the "15V" comes straight from an unregulated rectifier, e.g. from a mains transformer, then it's not really 15V even if an RMS voltmeter indicates so - the peaks are higher, and as the capacitor ages and loses capacitance and gains ESR, the ripple will only grow and accelerate the failure.
Ceramic capacitor performance strongly depends on the type of dielectric used. Dielectrics other than NP0 and C0G have varying degrees of capacitance change with applied voltage. Small AC voltages may even increase the capacitance slightly, only for it to "fall off a cliff" when the AC amplitude or DC component is increased. When using anything but NP0/C0G ceramics above a couple of volts DC, always measure their capacitance using a capacitance meter that can accommodate bias. If your RLC meter doesn't allow that - many types get damaged with even small DC bias - feed a square wave into the capacitor through a resistor, and derive the capacitance from the time constant observed on a scope. E.g. for a 9V application on a 10V capacitor, apply a 9V DC + 0.2Vpp square wave through a resistor, and observe the voltage on the oscilloscope. The RC time constant observed, with the known series resistance, lets you figure out the capacitance. Ceramics can fail "hard open" or "hard short". Mechanical stress causes cracking and that leads to open failures, but sometimes the layers misalign in such a way that the capacitor shorts out. There are also random insulation failures that result in shorts, but that's rare if the capacitors are otherwise not overstressed - i.e. when the operating voltage, ripple current, and operating temperature are all a bit away from the recommended operating range.
Unless you're trying to save single cents from a very high volume device, you'll not save any money by "economizing" on capacitors. In low volume professional applications, ample derating is more than worth it. For example, in high current LDO circuits I derate tantalums by 50-70% on the operating voltage vs. voltage rating, 50%-90% or more on ripple current, and keep them well away from the temperature limits. For aluminum electrolytics, I derate operating voltage by 20%, ripple current by 75% or more. For ceramics, the voltage derating is usually limited by very steep price increase once you're past a certain voltage-capacity product for a given dielectric category. For bulk decoupling I use physically large ceramics. For local decoupling in modern digital logic/CPU applications, the parasitics often matter more than the exact capacitance value, so a small case size is very beneficial, up to a point.
Depends on the capacitor type and environmental conditions.
For electrolytic caps, they are (generally) able to withstand twice the rated voltage for 1 or 2 seconds. So, having the voltage close to its rated shouldn't be a problem.
However...
Like in other components, a capacitor's ratings need to be de-rated with external conditions (e.g. temperature). This means that a capacitor's voltage rating might be lower for different temperatures. For example, an aluminium electrolytic capacitor's voltage rating will probably be lower at 80°C than that at 20°C..
For ceramic caps, as @TypeIA stated in their comment, the capacitance value drops significantly with the applied DC bias. The amount of drop depends on the class e.g. C0G/NP0 (Class I) drops less (more stable) than X7R (Class II) does.
C will get shorter lifetime with a proof or that'd be only myth?
For electrolytic capacitors, lifetime heavily depends on the temperature instead of the voltage. For aluminium electrolytic caps, 10°C difference is enough to halve (temp. increase) or double (temp. decrease) the estimated lifetime (For aluminium polymers it's 20°C).
One thing to consider here is the ESR-vs-voltage behaviour: As the voltage across an electrolytic capacitor increases, the ESR increases as well (this increase may or may not be low to ignore, though). If the capacitor is used as a ripple filter then, with higher DC voltage across the capacitor, the ripple current will dissipate more power across the ESR. This will bring an overall temperature rise which eventually results in reduced lifetime. With the same chemical properties (electrolyte and other materials), an aluminium electrolytic cap with higher rated voltage will quite possibly have lower ESR since the plates will be larger. This will bring less dissipation, less temperature rise, less lifetime reduction.
A 16V electrolytic cap on a 15V power supply will have shorter life than a 25V cap.
This is common in "cost optimized" designs which are not designed for quality nor reliability: since the thing won't last very long anyway, who cares.
It's more of a problem if it's used as smoothing cap after rectifiers, because mains voltage is not that accurate, and the output voltage of small transformers will change a lot depending on load. So if the cap voltage was chosen with the lowest possible safety margin based on the loaded output voltage of the transformer, but it isn't always loaded so voltage is often higher, then it won't last long. As it slowly dies, ripple on the DC will increase, until noticeable symptoms occur.
If you try this kind of stuff with a tantalum cap, it'll work fine until one day it decides it's had enough and immolates itself by fire in protest.
On a related note, some capacitors may need "reforming" before applying rated voltage, especially after long time disuse and storage. This might mostly apply to large electrolytics as used in DC bus links of VFDs. There is a reforming process:
https://library.e.abb.com/public/9f1befcd62ce445ec1257466003c02e2/3BFE64050629.pdf
https://www.qsl.net/g3oou/reform.html
So there may also be issues using a capacitor well below its rated voltage.
