Electrical – run a small kiln off a living room circuit with ext cord out to the yard.

I'm sorry, but from the tone of your question, and the fact that you even suggest plugging this motor into a regular 15A receptacle tells me you are so far over your head your only valid option is to hire a pro to wire this. This is NOT a simple DIY job and you cannot simply ask questions to get every little detail out of an internet message board to do this job properly and safely.

I'm sure I'll get negative marks for this answer but I stand by it firmly.

The other, and cheaper option is to unload this 3-phase motor and get a single-phase motor. I bet a 1hp single-phase motor is cheaper than the phase converter itself. 1HP is not that big and single-phase versions are readily available. You'll still need a new dedicated circuit for it but it'll save you quite a bit in the long run.

Circuit wire is kept relatively large because general purpose receptacles are just that: general purpose. You can plug a nightlight into them one moment and a kilowatt hairdryer the next.

In a fixture, luminaire, or appliance, on the other hand, the designers can use thinner wire (down to 18AWG for fixture wire as per NEC 402.6) as they know how much current will ever flow through there -- the appliance or fixture only needs a certain amount of current, never more (unless there's a fault, which is why you have fuses in appliances).