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? .