My battery is fully charged. I tried it on my cousins SGS3 and it showed 98%. I just tried what you said (hit the power button, let go then hold home and volume up buttons) and still no dice. I've also tried the normal way of getting into recovery (holding down all three power, volume up, and home buttons) with no avail.
It doesn't make sense to me (I'm fairly familiar with Android phones since the G1) since the custom reboot app is just that; a custom reboot. The mod should have downloaded into the Downloads folder which means nothing else other than my TouchWiz Jelly Bean CWM is in the root folder for recovery. Is there any way Dolphin Browser would've downloaded the mod into the root folder? If so, is there a way I can check via my PC?
EDIT: I went digging around the interwebs for more info on what's happened to my phone and I found that my phone also shows the QHSUSB_LOAD driver installation when plugged into my PC. From what I can gather, that seems to be a sign that I flashed an itnl ROM on my SGS3 and hardbricked my phone. It still doesn't make sense to me since I didn't flash anything after downloading the mod/custom reboot app.
EDIT2: I went to a local Sprint store to see if they could shed some light on it. I told them I downloaded a file and an app, went to reboot my phone and now it won't turn on (which is what I did, just didn't mention it was already rooted). He tried to figure out and came up with nothing since he said every browser app should download into its own folder and not the root folder (I think he knew my phone was rooted but didn't want to say anything. Cool guy). Before I get shamed for not taking responsibility for my actions, I offered to pay for a new phone replacement but the technician was kind enough to order a new one for free. While I will get a replacement phone Monday/Tuesday, I'd still like input as to what happened. It'd be helpful to know so others don't make whatever the same mistake I made.