Does a multi-outlet extension cord let me draw more total current than a single-outlet one?

No, the total amperage rating is the same whether the extension cord has one outlet or six. The wire gauge and length determine how much current it can safely carry — not the number of sockets. Plugging multiple high-draw appliances into a multi-outlet cord will trip a breaker or start a fire just as fast as using a single-outlet cord with a splitter. Most household extension cords are rated for 13 or 15 amps total, so treat it like a single pipe — the capacity doesn’t multiply just because there are more taps.