Are these problems related or coincidental??

I apologize for the multiple threads over the past month, but my problems are growing and I'm not sure if it is software or hardware.

It started with blue screens and it seemed to be either a thermal or PSU problem. I began to learn to live with it because I'd leave the computer on and some days I wouldn't get a blue screen. If I did, I'd turn it on a few times and it'd be fine. I'm currently testing the memory and it's been testing for over an hour (46% and 22%) with no errors.

Well two days ago I lost internet despite our laptops staying online. I spent over a half hour on the phone with the DSL provider service support and she concluded that my ethernet (Realtek PCIe Family Controller) is not working. ("Unidentified Network" "no valid IP" etc.) She suggested I buy a wireless adapter.

My question is, are all of these problems related? Could it be a virus (I did a full scan (one virus found: "EICAR Test--NOT virus" moved to chest) and a boot scan (one virus found: "Java:CVE-2013-1493-CI" moved to chest)

So to clarify, I get occasional blue screens (various different reasons) usually at or near the beginning of my using it and more recently, cannot connect to the internet (although my other computers are able to). I have also confirmed the ethernet cord is ok by using the same cord and successfully getting internet on my laptop. I just find it odd if these problems are not related. FYI this is an HP that's 4.5 yrs old. These are the other threads: http://forums.majorgeeks.com/showthread.php?t=286281 and http://forums.majorgeeks.com/showthread.php?t=286237

Thanks for your input!

 

I forgot to ask: If I decide to buy something to help me reconnect to the internet, would you recommend an ethernet card (wired) or a USB wireless adapter? Should I look for a certain specification? Thank you again.