Can I listen something with dongle without tuning this antenna?
Yes. Tuners can help with receiving with an antenna of an otherwise unsuitable length, but it is absolutely worth trying without them.
Here's a useful rule: Watch the spectrum and unplug the antenna. If this causes the visible signal level all across the spectrum to go down, then you had plenty of signal from the antenna and there is no need to improve it.
If it did not go down very much, then you'll get better reception by improving your antenna system — whether by an antenna of a different length, a tuner, a LNA, or another means.
Maybe RTL-SDR run from battery. Because of computer PSU ? Any suggestion for battery ? I need 5 V ? Any suggestion for accumulator (I will buy it later)
DO NOT wire a separate 5 V supply, whether battery or otherwise, into your USB connection unless you are willing to risk damaging the USB port/motherboard you plug it into by back-feeding power to it.
The minimum safe way to do this is to use a USB hub which accepts external power. The more-isolating way is to use a USB isolator to separate the power connections completely. USB isolators are unusual and therefore expensive equipment, however.
Another option is to get a computer that does not produce significant noise. Different models can vary widely.