Yes, but only to the extent it is not a perfect capacitor. A perfect capacitor is not capable of dissipating energy, which means it can't be heated by the current thru it since that would represent a loss.
Real capacitors have a spec called equivalent series resistance, or ESR. That is various physical factors lumped together. You can for most purposes think of the capcitor as ideal with a series resistor of the ESR value. A resistor dissipates power proportional to the square of the current thru it.
Real capacitors can get hot with sufficient current and can eventually fail as a result. Electrolytics are particularly susceptible to this. Not only is their ESR high relative to other cap technologies, but they are more sensitive to high temperatures. Some electrolytic capacitors are specifically designed for higher currents, such as the Panasonic FK series. Look at the FK series datasheet in comparison to other types and you will see the FK can tolerate higher ripple current. They are also somewhat physically bigger and more expensive than the other types.
Ceramics generally have very low ESR, but it's not zero. I once got a ceramic disk capacitor so hot it hurt to touch it, and that was at only 1 MHz. (Maybe the 10s of volts from a RF transmitter I was putting accross it had something to do with it :-) )
Dielectrics store energy in the D field by sloshing electrons around at least a little bit. Some also flip atoms between different energy states. Any time you move energy around, even at the scale of a few atoms, there is the opportunity to lose some of it as heat. Some ceramics lose less than others, and ceramics as a whole lose very little, but they all lose some.
Air capacitors don't lose any power in the dielectric since there isn't any (basically vacuum), but unless the leads and plates are superconducting there will be loss there as charges slosh back and forth.
No, there is no circuit being completed, hence no means for the "charge" to flow into the battery.
It appears that you expect a Coulomb static charge behavior to occur, such as in a Van De Graaf generator. That's static electricity, which albeit analogous to battery type "Direct Current" electronics, is not quite the same thing.
For one thing, batteries are not Coulombic charge storage devices: They are, at the simplest, reversible chemical reaction storage of energy resulting from ion flow caused by a potential difference. "No actual electrons are harmed in the battery process".
Also, though capacitors are technically charge storage devices, they would require a discharge path towards a lower absolute static charge potential point, for what you have asked, to occur. That's not the case here.
Best Answer
No. A capacitor only has one charge. The charge creates a voltage differential to the capacitor. From the positive pin, the voltage is negative towards the other pin. From the negative pin, the voltage is positive towards the other pin. What you suggest would mean that there would need to be 3 pins and two capacitors.
It would help a little if you would tell us what you are trying to do, exactly. If you are trying to make a negative voltage, that can be done by switching to which nets the pins of the capacitor is connected to. Let's say you need to generate -5 volts from +5 volts. You can charge the capacitor between the +5V net and ground, then disconnect the capacitor, and then connect the + pin of the capacitor to the ground. The - pin of the capacitor now has -5 volts, when compared to ground.
As you discharge the capacitor, the voltage at the - pin will change towards the ground; from -5 volts to -4 volts, from -4 volts to -3 volts and so on. So you will need to switch the way the capacitor is connected again, and charge it from between the +5 volts and ground. This system is called a "charge pump".