@CooperBlack 

Hi Cooper.  I assume since you tried all 3 WPS methods that the service is enabled.  What you can do is first clear out the wireless settings on the printer, thus ensuring it has forgotten the settings for the Netgear router.  While it shouldn't affect this its a good place to start since you are not able to get the service to work.  Another thing to look at on the router after attempting it is the router's system log.  See if there is anything in there about the handshake between the router and the printer failing.  A final option is you setup the printer manually using the SSID and password.  

@CooperBlack 

Yes the password is the same as a network key.  You would not configure this through the router though.  You would handle this the same way as connecting a phone or tablet to the network, from the client device itself.  If your printer is fairly new there should be a interface either on the printer itself or through its web interface or even the app.  For instance i was able to configure my HP printer via the HP Smart print app.  

@CooperBlack 

WiFi direct is basically a standard that allows two devices such as your phone and Printer to connect without a direct AP involved.  Similar to how Apple's Airprint service works.  This would need to be enabled on the printer and then the client device (phone) would need to be complatible. 

For the network security I would switch to WPA2-PSK with AES.  Its far more secure than WPA is.  That simple fix may correct the situation,  even if it was working with Netgear its one of those trial and error things.  Finally if this is the only device having issues we would suggest contacting Samsung to help walk through the process.  I hate saying things like that but since we do not have direct client device support and do not have any familiarity with these printers that might be the best option and accomplish the goal in the quickest amount of time.