No, don't cut the plug off the pump. Install a new electrical outlet near the location of the pump and plug it in. Unless you're saying the pump cord will reach an existing electrical outlet. If so, my bad, I didn't realize you had an outlet so conveniently close.
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You can, if you want, just lay the wires on the surface of the ground. It's just not a good idea to bury them without a protective sleeve or conduit. If your concerned the exposed wires would be unsightly or a tripping hazard, then yes, your idea is a good one. Lay a large enough PVC electrical conduit to accommodate all your wires. Make sure the conduit is large enough to pull the plugs through. You can either bury the conduit or run it on the surface. If you leave the ends of the conduit open, it would be a good idea to use some sort of sealant to prevent water from filling the conduit. There's a stuff called duct seal that we use. It's sort of like clay and doesn't harden so you can still have access in case you need to pull your wires back out. Everything you need can be bought at a hardware store or big box stores such as Lowe's or Home Depot. It won't cost much.
By the way Craig, you are a lucky man to have a wife so interested and willing to take on this project like this. Mine is the same way. I declared her a saint years ago! 
Yup, what he said.