You can plug a surge protector into an extension cord, but watch the gauge.
Yes, but don’t make a habit of it and pay attention to the wire thickness.
The real risk isn’t the surge protector itself — it’s daisy-chaining power strips or using a thin extension cord that can’t handle the load. If you’re using a heavy-duty 12 or 14 gauge cord meant for power tools, you’re fine. If it’s one of those skinny light-duty cords from the dollar store, that’s where fires start.
The surge protector needs to be plugged directly into a wall outlet to do its job fully — the extra length can reduce its ability to clamp a surge. But for temporary setups, like running a power strip to a desk across the room, the main thing is making sure the extension cord is rated for the total wattage you’re pulling.
Bottom line: thick cord, short run, and unplug it when not in use. This is not where you save five bucks.