Can batteries be charged and discharged at the same time? And do they wear down faster when doing so?
Your confusion is not about batteries but a basic issue with current.

No, a battery can't be charged and discharged at the same time.  If a battery is connected to a charger delivering 1 A and a load drawing 3 A, then the battery will be discharged at 2 A.  There is no simultaneous charging and discharging going on.

Draw out the circuit and follow the currents.  You can conceptualize the above example as 1 A charging the battery and 3 A discharging it, but the battery sees the sum.  Again, draw a diagram and it should be more clear.  Handwaving makes everything difficult to understand.

Added:

Your diagram should make it clear.  You can't have current going thru the battery both directions simultaneously.  You can think of it conceptually as being two currents, but the battery will see the sum of the pink and green currents in your diagram.