If i play Promo Eevee onto my bench

If i play Promo Eevee onto my bench, then evolve another pokemon on my bench, can I use Eevee's power to look for an evolution and evolve it, even though I can't evolve a Pokemon the same turn I put it down?

Yes. It counts as evolving Eevee, but you are not actually evolving him. There is a difference.