How to lay sheet vinyl flooring in a complex, larger space. Our DIY kitchen reno slate-look vinyl

After we learned that the vinyl flooring in our kitchen had asbestos in the glue, we decided not to rip it out and lay oak flooring to match the rest of the house. I considered tile, but tile is tricky in a kitchen... it's hard underfoot, hard when you drop things on it or kids fall on it, has grout lines that can be annoying to clean, and takes time to lay. But I do like the look of tile--I thought about terra cotta or slate as a solid but varied, rich color.

Then I thought about vinyl. I...