This is what I know: the charging of a capacitor from a dc source
happens instantaneously. Since there is no series resistor to limit
the current, then what actually prevents the current to become
infinite and burn the capacitor at charging time?
Alas, if a capacitor will really "burn" at high currents, it will burn regardless of which way the current is going.
On the opposite is the discharging. If a charged capacitor is shorted,
it will burn. Otherwise said, the discharge can not happen as fast as
the charging; why the capacitor can charge instantaneously but it can
not discharge as fast?
I've never seen this happen. A single charge or discharge cycle will not realistically destroy a capacitor. I suppose there are might (might, I say) be caps which self-destruct in the event of a single event, but frankly I've never run across one in 50 years of messing around with electronics.
What can happen is that, if a capacitor is repeatedly charged and discharged at high frequency, some types can overheat and blow/burn up. In this case, you use a capacitor with a lower internal resistance (called ESR, short for "Equivalent Series Resistance"). For any capacitor, internal power is produced by the internal resistance of the cap, which produces a power related to the square of the current divided by the resistance, and if you make a thicker internal structure the resistance decreases. (This gets complicated at high frequencies due to phase shifts caused by both capacitive and inductive effects, but you don't need to worry about those just yet.)