Good news! Problem Solved!
In the end the solution came from home after all, but thank you for your help and advice WindowsSniper as it pointed me in the right direction. My sister ended up buying the same laptop, and we noticed that hers works just fine with headphones. Funny part was, the Realtek Control panel on hers looked nothing like the screenshots I had seen; turns out Dell uses their own customized OEM version of the software, the control panel has their logo all over it and that's why my taskbar icon is black-and-white. I knew a lot of laptops came with OEM drivers, but I didn't think that would be preventing the download from Realtek's site from installing properly.
In any case, after several times reinstalling Dell's version from the driver CD with no effect, and getting that same error message "Rundll32.exe has stopped working" I figured there might be a corrupted file on there or my laptop because it obviously was not installing/running properly. As a second thought, it may have been conflicting with the Soundblaster X-fi MB software if that didn't install properly.
Solution: I manually uninstalled the entire Realtek package, making sure to clean out everything. I then reinstalled the SoundBlaster software, using the full installation settings. After that, I picked up the Driver CD that came with my sister's laptop and used that to reinstall Realtek - this time no error message, and the control panel / headphone detection is working fine now. I'm still running an older version of the driver, I'll try to update that later through the device manager but at least I know this version of Realtek is working
I might be wrong, but my suspicions are:
Either SoundBlaster didn't install properly and was conflicting with Realtek, or the download I got from Dell didn't install properly, causing the original problem.
A tiny scratch on the disk says my Driver + Utilities CD was not reinstalling Realtek properly. Downloaded versions weren't working right because I needed Dell's OEM version.
Again, thanks for all the advice; I'm just glad I don't have to send this back in