Can two normal batteries be connected by series and parallel, at fraction of second apart, so as to charge and discharge at the same time?
There are components available to switch batteries between series and parallel.

However, there are no problems for which that is the best solution.

Matching voltages by dynamic series/parallel connection is far better done with a buck or boost SMPS converter. The effect of charging/discharging 'at the same time' is better done by powering the load directly, and making up the balance of the power flow to or from the battery, using voltage conversion as required.