Alright, here is the progress that was made after I read your post:
I went into Safe Mode with Networking, and I had the same problems as I had in the computer's normal mode. Those problems were as follows:
1.) Both exeHelper and DDS gave me an error message that said, "The parameter is incorrect."
2.) Win32kDiag gave me an error message that said, "…is not a valid Win32 application." (Same error message I get when I try to download an executable file.)
3.) When I double click the GMER.zip download, a folder comes up that is empty, hence there is nothing to extract.
I do have access to another computer, so I was able to save 2 of the programs to my flash drive: Win32kDiag and GMER. (I will post those logs here.) However, I had no success with the other 2 programs you listed. When I tried to download exeHelper, a threat was detected that said the file was infected with a virus. I don't remember what happened with DDS, but I wasn't able to download that one either.
Now, when I run GMER on this computer, it starts scanning automatically and does find some things. (I will also post this log that has nothing unchecked.) However, in your post, you told me to make sure a few things were unchecked, so I unchecked those things and ran the scan again. When I do this scan with the items unchecked, the program stops running and gives me an error message that says, "gmer.exe has stopped working. A problem caused the program to stop running correctly. Windows will close the program and notify you if a solution is available." So, that's about where I'm at right now. I have listed the logs below.
Another thing I just noticed: In your post, you also said that when the Win32kDiag log was finished, it would say, "Finished! Press any key to exit…" I had the program running for a while, and I never received that message. I'll post both of the logs anyway. Maybe you can find something that hints to what is wrong with my computer.

I appreciate your help so much.
Here is the Win32kDiag.txt log that I have:
Running from: C:\Users\[removed]\Desktop\Win32kDiag.exe
Log file at : C:\Users\HP_Administrator\Desktop\Win32kDiag.txt
WARNING: Could not get backup privileges!
Searching 'C:\Windows'…
Cannot access: C:\Windows\System32\LogFiles\WMI\RtBackup\EtwRTDiagLog.etl
Here is the GMER log that had all items on the right-hand side checked:
GMER 1.0.15.15252 -
http://www.gmer.net
Rootkit quick scan 2009-11-23 22:15:57
Windows 6.0.6002 Service Pack 2
Running: gmer.exe; Driver: C:\Users\HP_ADM~1\AppData\Local\Temp\kwgdafod.sys
—- Devices - GMER 1.0.15 —-
AttachedDevice \FileSystem\fastfat \Fat fltmgr.sys (Microsoft Filesystem Filter Manager/Microsoft Corporation)
AttachedDevice \Driver\tdx \Device\Ip avgtdix.sys (AVG Network connection watcher/AVG Technologies CZ, s.r.o.)
AttachedDevice \Driver\tdx \Device\Tcp avgtdix.sys (AVG Network connection watcher/AVG Technologies CZ, s.r.o.)
AttachedDevice \Driver\tdx \Device\Udp avgtdix.sys (AVG Network connection watcher/AVG Technologies CZ, s.r.o.)
AttachedDevice \Driver\tdx \Device\RawIp avgtdix.sys (AVG Network connection watcher/AVG Technologies CZ, s.r.o.)
—- EOF - GMER 1.0.15 —-