Your thinking of this in the wrong way my brother. A switch is not a router. they simply bounce communication between sources in the local network. The internet is seen as an outside network by a router and it uses a dns server passthrough to allow you to surf the net but stay on the local network at the same time. Your internet connection can only be 1 IP adress so it would be impossible for you to have more than 1 pc using it at a time without the router to handle traffic between the outside network and inside network effectively making all your pc's look like a single IP on the internet. If I am correct you have a pc somewhere with a wireless connection to a router and you want to use its wired NIC at the same time to share internet to other computers on a switch yes? If so you do want to enable connection sharing and make sure you have the NIC card enabled in the device manager. If all those things are done then you are half way there. If you can connect other computers to the switch and gain local access only then the issue lies in the path way between the inside network and outside network. This means your PC will become the router and it has to now do the traffic handling. When you connect a normal computer to a wifi router it will send an auto config file after your password has been verified which then gives your computer its IP address on the router, The primary and secondary DNS servers for internet, and any specific port info. In this case that file didnt get sent because it doesnt exist. You have to change those settings manually. In one of the computers hooked to the switch you need to go into the adapter properties and click on TC/IPv4 and manually assign it a DNS server. The IP and gateway can remain automaic but the primary and secondary dns must be the same as those used by your ISP. They can be found by checking the connection details on the internet handling modem/pc.
Another method around this would be to buy an access point and connect it to your main router wirelessly and then wire that AP to the switch. This would be the easiest and would also give better performance in internet and file handling. It is effectively the same things as what your attempting to do now but it requires no further setup than connecting the AP to the switch and also eliminates the need for an internet handling pc to always be running. ASUS makes some cheap but good AP/Repeater/Routers called the RT-N12. They cost about $40 canadian and are easy as pie to setup and use.
Yes exactly. The N12 in this case will act as a wireless acess point for your switch effectively forwarding its ip addresses wirelessly to the primary routers DHCP server. This can also be called client bridging. The computers hooked to the switch will function at gigabit speeds while communicating with eachother and will be restricted to wifi speed when connecting to any other pc not wired into that switch. In this case the internet will most likely be the only transmission over wifi and will be more than sufficient unless ofcourse you are running some sort of T3 or high speed fibre internet connection.
To change the IPv4 settings on your switch attached PC if you are running windows is to open network and sharing center and click change adapter settings. You should see your connection there, double click it and hit properties. Double click TC/IPv4. This is where you can define a primary and secondary dns.
The DNS info can be found by logging into your modem and looking at the connection settings.