In plug-in hybirds: If gas engine is used for weeks and no battery charge, would that damage or affect the lifespan of the electric system?

No. All hybrids have regenerative braking which stores what would usually be wasted energy into the main traction battery.

They also monitor the traction battery and use the internal combustion engine to partially recharge the traction battery if the charge drops low enough to cause potential harm.

The vast majority of hybrids are not plug-in models. But plug-in hybrids have larger traction batteries and can take advantage of an external electricity supply to charge the battery without needing to use the car internal combustion engine and brakes.

Hybrid traction batteries last a very long time as they are never allowed to be truly fully-charged nor fully-depleted. Typically limits of about 80% maximum charge and 20% minimum charge are applied by the vehicle firmware. That’s why the few Toyota Prius Mk1 cars which are still around (from around 1997 to 2003) usually still have their original traction batteries. They usually get scrapped just because the rest of the car has worn out.

The charging of the battery would be done by the car itself from the power of the engine running its generator for the battery. The battery has two ways of being charged- plug in or while driving. Plug-in charging doesn't cost any gasoline.

Doesn't it charge the batteries when the gas engine is on?

Can just tell you that our friends just traded a hybrid . . . was 11 years old . . . on theirs, the gas engine only came on when batteries were getting low . . . . they noticed it was coming on more often, ie batteries were not holding charge as well. So - they traded it in - but 11 years is longer than they expected the batteries to last !