UPDATE TO MY LAST POST -
The steps I listed above only worked for my iPhone 5c. For my sister's iPhone 5c, the steps I described did not, in fact, work. We replaced the cable for her phone, and still iTunes was crashing as soon as I would plug her phone in.
What the problem was...
She had used a program on Windows called 'Sync iOS', to add some ringtones and music to her phone. Apparently this was causing the issue with the iTunes crash. Since I could not use iTunes to back up her data, I actually used the program 'Sync iOS' to back up her contacts, apps, photos and videos to her desktop. Then, in order to get the iPhone to connect to iTunes without crashing, I had to do a Factory Reset on the Phone.
Once I Factory Reset the phone (careful, as this will erase all user content and settings) upon the next boot, I was able to connect to iTunes and update the phone. I then used Dropbox to replace the contact list, photos, and videos onto her phone. It was a bit of a headache, but I figured I would post this here to help anyone in the future.