Hi and Welcome to Seven Forums,
First off, make sure that you have this type of connection: Modem>Router>Switch>Wired Computers/Devices
Do I have to assign static IPs to all devices?
Yes you may, but you may leave DHCP or Auto IP on your computers and network devices that will need access to the Internet. You may assign Static IP to the ones that won't need the internet connection but will be part of your Homegroup or Workgroup to share and access network resources such as files, printer, etc. Assign a Static IP only on IP Address and Subnet Mask,
the Gateway and DNS Servers will be left blank.
Do I have to create a new subnet mask so my PC can differentiate between the two networks?
There's no nees, they will all be on the same Subnet, easier to configure this way and they will all be in one network.
Do I have to set a new default gateway separate from the wireless router?
See the first answer. No need.
Do I require a DNS server?
Again, please see the first answer.
This is all you need and you'll be good to go.