" Do they literally cover every hex of the Attacker's side of the battlefield from start to turn three and then the entire battlefield in turns 4-5?"
They affect every unit in that area, yes. The individual Tukayyid Boar Spiders are not tracked. They are an environmental effect, not units, rules-wise. The stampede covers an area, has effects, and when the stampede is done, is no longer has any effects.
There's no non-stampeding Boar Spiders. Once the stampede is over, the boar spiders are gone.
No, they can't restart.