Will capacitor in high pass filter pass high frequencies while it is blocking low frequencies?
And at low frequencies and DC currents the cappacitor gets fully charged very early at the beginning of the cycle and acting afterward as an open circuit until the voltage drops in the second half cycle below the capacitor voltage when the capacitor starts to discharge.

No, you have been given a completely unhelpful picture of how a capacitor behaves.
If you apply DC to a capacitor, then it takes an initial charging current while its voltage is changing. Once the capacitor has charged to the supply voltage, no further current is drawn. Both of these states are summarised by the equation I = C*dV/dt, the capacitor draws a current proportional to its capacitance, and the rate of change of voltage across it. Initially, there's a large dV/dt, so a high current. At the end dV/dt is zero, so the current is zero.
Exactly the same equation can be used to predict its behaviour with AC.
High frequency AC has a large dV/dt, so the capacitor draws a large current, and appears as a relatively low impedance. Low frequency AC has a lower dV/dt, so the capacitor behaves as a higher impedance.
In a linear circuit, voltages and currents can be superposed without interference. This means you can analyse the response of the filter components separately to any given input signal. If you supply all of those signals at the same time, its response will just be the sum of those responses. In a highpass filter with a series capacitor, high frequency signals will find a low series impedance and be coupled to the output well, low frequency signals will find a high impedance and be poorly coupled to the output.