Tiny swarms and weapons with multiple damage types

Yes.

Damage from the same source with multiple damage types (such as a grimlock's Spiked Bone Club attack, or the flame strike spell) is all inflicted at once. Nothing in the rules for Damage Types (PHB, p. 196) indicates that a given effect is restricted to a single type of damage. The wording for these effects doesn't separate the damage types into different sources:

Spiked Bone Club, MM p. 175

Hit: 5 (1d4 + 3) bludgeoning damage plus 2 (1d4) piercing damage.

Flame Strike, PHB p. 242

A creature takes 4d6 fire damage and 4d6 radiant damage on a failed save ...

If flame strike's damage types were supposed to be delivered by different sources, then they wouldn't be conditional on the same (singular) saving throw. Likewise, the Spiked Bone Club's damage is the effect of the (singular) Hit.

To further support this, the optional Vitality rules (presented in this Unearthed Arcana article) use this language:

Whenever a character takes 10 or more damage from an attack or effect, the character loses vitality.

This makes it clear that a single attack's total damage, regardless of its constituent types, is what matters. The Massive Damage and Instant Death rules have the same flavor and can be construed to work in the same way.

Of course, resistances and vulnerabilities may apply separately to each damage type that an attack inflicts. A monster that is resistant to fire but vulnerable to radiant damage that fails its save versus a flame strike will take an average of (14 / 2) + (14 * 2) = 35 damage from the spell as a single damage source.

This rule comes from D&D 3.5, but was not carried over to Pathfinder. The rule is:

A lit torch swung as an improvised weapon deals 1d3 points of fire damage per hit.

You can find this in the "Vulnerabilities of Swarms" section of the d20srd Swarm entry, next to other text that talks about how swarms are especially vulnerable to energy attacks in general, like Flaming weapons. In PF, swarms take extra damage from area attacks, but have no vulnerability to energy attacks.