Generally, yes, they are the same. The term "lead storage battery" is a bit incorrect, but it conveys the idea that the battery in question is a lead-acid storage battery. These storage batteries are...
They have lead plates. When they wear out is when you replace the battery.
It is a battery which, as you may have guessed, used lead and an acid to store an electrical charge. Most car/truck batteries are of this type.
Yes, they contain sulfuric acid.