I checked my Device Manager, and it showed two instances of my Realtek Ethernet driver. One of them was from Realtek, but the other one was shown as being from SeriousBit. I had installed SeriousBit NetBalancer awhile back, and it may have been around the time that I first started noticing the problem. I wonder why it installed a duplicated instance of my driver.
I went ahead and uninstalled NetBalancer just to see if that would solve the problem. I also decided to uninstall my network driver and reinstall a fresh copy of the original one to be safe. I had to boot in Safe Mode to do so (In regular mode, the progress bar got stuck). When I tried reinstalling the original driver, I got an error message saying, "This version is for Windows Vista only. For other versions of Windows, please use the other platform ethernet driver installer." This was the driver I had downloaded specifically for Windows 7, so I'm not sure why I was getting that message.
After booting back up into regular mode, Windows automatically tried to reinstall the missing ethernet driver on its own. It then reported the following error in the Action Center:
Description
Windows was able to successfully install device driver software, but the driver software encountered a problem when it tried to run. The problem code is 1.
Problem signature
Problem Event Name: PnPDeviceProblemCode
Architecture: x64
Hardware Id: PCI\VEN_10EC&DEV_8168&SUBSYS_3610103C&REV_02
Setup class GUID: {4d36e972-e325-11ce-bfc1-08002be10318}
PnP problem code: 00000001
Driver name: Rt64win7.sys
Driver version: 7.2.1125.2008
Driver date: 03-02-2009
OS Version: 6.1.7601.2.1.0.256.1
Locale ID: 1033
Files that help describe the problem
DMI9B06.tmp.log.xml
oem23.inf
I can upload the xml and inf files if anyone wants to take a look at them.
I tested my computer to see if it would Standby and Hibernate successfully, which it did. So hopefully, the original problem is solved. My Internet seems to be working, so I'm not sure if I really do have a problem with the way my driver was installed or not. I know I'm using a driver back from 2009, but whenever I try to update it in the Device Manager, it says I already have the latest one. I checked the HP website, and they actually show a driver from 2008. I checked the Realtek website, but I was unsure if the latest driver they had would be compatible with my specific laptop. Some of the specs looked a little different, but I may not have been looking in the right place.
Thanks again everyone for the help.