My electric chainsaw's motor spins if I pull the chain. Is that normal or am I hurting it?

What you're feeling is pretty normal for most electric tools with brushed motors; you're just back-driving the gearing and the motor's armature. There's no hydraulic or compression lock like an engine, so a gentle turn is expected and won't damage it. The real consequence isn't damage from spinning it, but the risk of accidental startup if it's plugged in or the brake isn't engaged. Always unplug an electric chainsaw before any chain manipulation or inspection. If it feels exceptionally stiff or grinds, then you've got a problem that needs a pro to diagnose the motor or gear assembly.