Can I tap Sliver Hive to pay for Hive Stirrings?
No, you cannot use the coloured mana from Sliver Hive to cast Hive Stirrings.

Sliver hive says 


  Add one mana of any color to your mana pool. Spend this mana only to cast a Sliver spell.


But Hive Stirrings is not a Sliver spell, it is only a Sorcery. This means you cannot use the coloured mana from Sliver hive to pay for the spell.

You can use the mana from Sliver Hive to play Creature - Sliver spells, or to play tribal sliver spells. There aren't any tribal sliver spells I can see, but you could play a changeling one such as Wings of Velis Vel.