Is flow attached at fixed propeller blade at every RPM / speed conditions?
No, flow separation far away from the intended operating point is common.
Witness racing planes. Before Ratier made the first useable variable-pitch propellers in 1932, high speed aircraft had abysmally low propeller efficiencies at low speed, so much that world records were set with floatplanes (which had virtually unlimited &quot;airfields&quot; at their disposal). Same with small racing boats: To get over the hump (transition from buoyancy to planing), drivers need to lean forward and to patiently gain speed with their high-pitch propellers providing only marginal thrust due to flow separation.
The rotating movement of propeller blades helps to accelerate the boundary layer outwards, so flow separation on propellers happens much later (at more adverse pressure gradients) than on wings. Regardless, if blade angle and flow direction at the blade don't match well enough, separation is unavoidable.