Printing the self designed PCB

Set the Net value of the plane to the same as the Net value of the ground. It'll merge them all together, then.

There is almost no reason to design a one-layer board these days. The entry level service for any board manufacturer will be two-sided boards with plated through holes. The only reason I can conceive of for doing a one-layer board is if you're going to etch it yourself and want to avoid connecting traces on the front to traces on the back.

I haven't used Fritzing's (what appears to be) magic for converting a wiring diagram into a PCB, but at OSHPark pricing it's worth a shot I'd say. It will either work or not depending on how sensitive your interfaces are to things like noise and trace lengths and such. But if you're wiring it up willy-nilly on a breadboard and it works, the chances of it being dumb luck are pretty low. It will probably work with whatever auto-routing Fritzing is doing behind the scenes just as well.

For the record, though, this is not the usual way PCBs are designed. Usually you'll do Schematic Entry, then you'll do Place and Route (that is placement of components on your board, and routing of traces between them). Many experienced designers will tell you that you should do the routing manually and maybe use the auto-router as an indicator of whether your component placement is OK and iterate. Once you're satisfied with your layout, you would generate Gerber files, which are vector artwork format for manufacturing. You get one file per layer, and the layers are typically:

1. Top copper
2. Top silkscreen
3. Top soldermask
4. Bottom copper
5. Bottom silkscreen (optional)
6. Bottom soldermask
7. Board outline
8. Drill file

You might also have a Milling layer, and if you're doing surface mount, you'll generate a Paste layer for the top and bottom as necessary to have a stencil created.