Does a Car Battery with a dead cell cause failures of alternators?

The only thing I will add to the good advice you’ve been given is that periodic testing is a good idea and should be performed a certain way.

The battery should be placed on a charger for at least 30 minutes (preferably an hour) and a load applied to the battery with a carbon pile.
The load should be approx. equal to what the normal starter draw is on a warmed up engine. NOT the intial current surge, but the normal steady amperage draw.
(125 amp draw, so 375 amp load applied)
This load should be applied for 15 seconds and the battery should maintain a voltage reading of 10.2 at a minimum. If it drops below 10 you’ve got a weak battery and if it drops to 9 volts, then to the boneyard it goes.