Network connection signal Excellent, but no internet

Short version of a long story...my son's commuter started acting up, shutting itself down after being powered up for as little as 10 minutes, for no apparent reason. He brought it to me to look at, first step was to open the back and clean out the dust (wasn't much to be concerned about). Powered right up for me and stayed on. I have wireless internet in my home, his computer uses a Linksys wireless USB adapter. Was able to find signal and get connected to my network, but no matter what it would not connect to the internet. Tried different things for several hours. Was not able to check for virus or spyware (for various reasons), so I decided to just reformat the HD. I am pretty computer savvy, but this one has me stumped. Did the reformat, again installed the Linksys adapter, found and connected to the network signal, but could not connect to the internet. Now here is the real kicker - my son bought this computer from someone on Craigslist.....it is a Gateway, but they removed the sticker with the model number, and Properties in My Computer does not provide the info either. There are a few drivers missing (Ethernet Controller, SM Bus, Video Controller, etc.) but without the Gateway model # I can't find the right ones (unless I use generic??). Could work around the wireless thing if I could get the Ethernet driver installed so I can hardwire to the internet temporarily. Any suggestions for a work-around or how I can find out the actual model number of the computer other that the two options already mentioned? Any help would be greatly appreciated!! BTW - the computer ran all night for me just fine, but this morning shut itself off on two different occasions.

 

What is the OS ?

There is another computer on the network using wireless and that works fine? On a computer that is working, go to the command prompt, run the command ipconfig /all and copy/paste that here. Do the same with the one that isn't connecting to the net and it's fine to just type in the entries for DHCP (yes/no) autoconfiguration enabled (yes no) IPv4, subnet mask, gateway, dns servers.

While you are in the command prompt, if you run pings do you get any answers?
ping 4.2.2.2
ping google.com
ping (the ip of the gateway that shows in a computer that does get online, without the ())