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I'm not sure what you learned about DNS servers but I explain it like this:
The internet doesn't understand words. It only knows numbers. Every webpage has an "address" that is a series of numbers that points to where a webpage is located. When you type words in your browser (like www.whatthetech.com) your browser sends those words to a DNS server that then tells your computer what the numbers are that will give you that website. It kind of like a phone book. You look up a name and it gives you a number to reach the party you are looking for. The nice thing about this system is that websites get moved all of the time. However, the DNS server get repopulated with the new location so you, as a user, never have to know that. The DNS server will report back the correct number within an hour or two of the change being made. Most people setup the DNS server of their ISP as the default (in your case Time Warner), but often the public DNS servers like Google or OpenDNS are faster.
I don't know of any adverse effects that changing DNS servers should cause you.
To back track a little, purple is not a network. It is the name of your computer. You can see it in several of the logs you provided. For example, an excerpt from FRST:
Scan result of Farbar Recovery Scan Tool (FRST.txt) (x64) Version: 28-06-2014 02
Ran by l (administrator) on PURPLE on 30-06-2014 08:23:49
Running from C:\Users\l\Desktop
Platform: Windows 8 (X64) OS Language: English (United States)
Internet Explorer Version 10
Boot Mode: Normal
the rest of the "weird" line - wi.rr.com is the time warner network (also known as RoadRunner). The whole line is just saying that the computer named purple is connected to the Time Warner network - which is correct as your computer is named purple and you gain access to the internet through your internet service provider... Time Warner.
I'm not convinced that this will resolve everything... I just want to try it. If issues still continue, we will reset your MBR. TDSSKiller didn't find a rootkit... but there could still be something there that is messing you up.
