In the absence of overt allergic symptoms, does a high ige level (600; normal 0-100) indicate a protozoal infection like trichomoniasis or babesiosis?

Usually not: First of all, the old biology-class story that "IgE fights parasites" isn't true; it's mostly about worms, not the protozoa you mention. Usually we never find the cause for a high IgE, though some folks with bad allergies run high levels and there are a handful of uncommon diseases where they run high. Best wishes.

Elevated IgE: Mainly indicates allergic disease or parasitic infestation, not sure if trichomonas is an infestation. But why was the test ordered in the first place?