Just wondering the proper way to finish this project:
This is probably something you should have thought about before placing the new flooring :)
However you should probably put a door sill in. As for material there are various options usually stone, wood or metal. You'll have to choose what best suits your home. If you want the drop to be gradual you should get an angled sill. Most likely you will end up having to cut into the newly laid floor a bit to accommodate for the door sill.
Finish end at step problem. You said, " I was more thinking finishing it with a stair nose piece."
Yes, I do it that way. They come 2" x2 with screws to install and will hold up to traffic. Also...
There are rubbery vinyl edge products that come 4 inches deep, and with a lip down. This will look more professional. Black or brown colors. Glue made for it holds it permanently in place. Both styles come large and need cut to fit. Hope this helps.