Do pilots turn off the engines during de-icing?
Generally, it's not required to shut the engines down.
Sometimes you're deiced at the gate or just off the gate, and in those cases you've not started the engines yet. Or if it's necessary to remove ice on the bottom of the wings or from the bottom/front of the (extended) flaps, those would be cases when you'd shut the engines down for deicing. But most commonly, you taxi to a de-ice pad (parking area), leave the engines at idle with the bleed valves closed (to avoid getting deice fluid into the air conditioning system - even a trace of it creates kind of a stink), and deice with engines running.
The engine cowl is deiced NOT with fluid but thermally, tapping bleed air (upstream of the bleed shutoff valves, so it's always available) to heat them. Thus the fluid is not directed at the engines, but is intentionally kept away from them.