"printer Settings Could Not Be Saved. Error Code (0x0000000001) Or Something"

I am running Windows 7 and yesterday my wife bought an HP 3512 printer. It has wireless capabilities, so she thought it would be nice to be able to print from other computers or tablets or phones in our house.I originally had issues even getting it to connect via wireless but it is now connected to my router, has an IP address, and if I log into my router, I see its connection. The printer works fine if I have it plugged in via USB. But as soon as I unplug the cable, the printer goes offline. Now, I assumed that by wireless printer, that meant I would not even need it hooked up to my desktop. Maybe I am wrong on that though.But my main issue is I cannot get the printer shared with the homegroup. If I run the printer troubleshooting, it even tells me that the printer is not shared with the homegroup and I should try to have it shard as administrator. So I click that box then it tells me it found but could not fix the issue. It's still not shared. If I go to the printer, go to properties, then sharing, if I try to click share and apply changes, I get an error saying "printer settings could not be saved. error code (0x0000000001) or something. It simply will not go onto the homegroup. If I add it to the homegroup to be shared, would that allow me to move the printer off of my desk and unplug the USB and still be able to print?Also, still regarding Homegroups. My computer is connected via wire to my FIOS router. A laptop in the other room however, connects wireless. They somehow end up in different networks and cannot join the same Homegroup. Both run Windows 7. I have looked and cannot find why they would be on different networks. They were not always this way, I use to be able to see the laptop on my desktop but no longer can.