Swarm Traits here
http://paizo.com/pathfinderRPG/prd/bestiary/creatureTypes.html:
"Swarm Traits: A swarm has no clear front or back and no discernable anatomy, so it is not subject to critical hits or flanking. A swarm made up of Tiny creatures takes half damage from slashing and piercing weapons. A swarm composed of Fine or Diminutive creatures is immune to all weapon damage."
My reading of that, and logic, says that bludgeoning would do full damage to Tiny creatures (it would be even easier to use a bat/club to damage a swarm than to use a longsword or rapier).
However, the spider swarm is Diminutive, and therefore immune to all weapon damage, and that would include unarmed strikes/stomping.
http://www.d20pfsrd.com/equipment/weapons/weapon-descriptions/strike-unarme d/
"The damage from an unarmed strike is considered weapon damage for the purposes of effects that give you a bonus on weapon damage rolls."
I'm not sure I buy the logic...if unarmed strike was supposed to be weapon damage, why wouldn't it just read, "The damage from an unarmed strike is considered weapon damage." rather than going on to say "...for the purposes of effects that give you a bonus..."
Unarmed strikes are weapons, as are natural weapons such as claws, bite, and so on. The swarm rules have no need to restate this rule.