1)How can a current flow from the battery to any one of capacitors to
get it charged without a closed loop??
First of all, if the circuit shown in your diagram has existed for a long time there will be no current flowing. That is because the voltages across the capacitors will not be changing in time. The current in a capacitor is given by
If the voltage across the capacitor is not changing in time, there will be no current. The capacitor looks like an open circuit in dc circuits after any transients (time changing voltages and currents) have died out, which is the situation after a long time.
But if the capacitors originally start out not charged, or not fully charged to the potential of a battery, current will flow once switched onto a voltage source. The battery pulls charge from one plate and pushes that same charge onto the other plate, creating a net positive charge on one and negative charge on the other. To do this the battery does work. The energy of that work is stored as electrical potential energy in the electric field between the plates as well as dissipated as heat in resistance that is always present in a circuit.
It is this process of pushing and pulling charges that results in current flowing in the conductors connecting the capacitors to the batteries, even though though charge does not actually move between the plates within the capacitor.
2)In capacitors C¹ and C² the charges are supposed to be equal as many
textbooks regards, but why and how could it be??
Since the capacitors are in series, when the capacitors are switched onto the batteries the same current (charge per unit time) flows to each capacitor. Since the same charge flows per unit time to each capacitor, the plates of the series capacitors acquire the same amount of charge, regardless of the value of capacitance. And that charge will be
So the voltages and charge can be calculated.
Hope this helps.