Interesting New Year's (and first) post...
First, you've fallen for the hype. There is no such thing as a 240Hz HDMI cable. The cable doesn't "know" anything about refresh rates. In fact, as you found out, the input signal is 60Hz and not 240Hz. So even if the cable had something to do with it (and it doesn't), the update rate (refresh rate) is coming into the TV at 60Hz.
That's where your problem is however. LCD TVs basically take the picture that is being updated at 60 times a second and "create" the missing 180 frames a second to make a picture that updates at 240 frames a second. This attempts to make the picture smoother since LCD TVs (even those with LED backlights) are slow in changing pictures compared to other types of TVs.
In addition, many TVs include settings for motion compensation. This is an attempt by TV manufacturers to compensate for some of the slowness by predicting how the picture will change and adjusting the picture accordingly.
You are not the first to notice that the combination of these two items causes artifacts on the picture. Many times these artifacts appear to be comet-like blobs around fast moving items and other things you wouldn't normally want to see.
A third possibility is that you have the sharpness setting too high on the TV. This can cause a ring around objects. The store settings / default settings for contrast, brightness and sharpness are all inappropriate for a home setting.
So, what to do?
1) Reduce your update rate to 120Hz.
2) Lower the amount of motion compensation to its lowest setting number. If numbers aren't available, set to Low or Off.
3) Properly adjust your TV settings. Unfortunately without your model number I can't be more specific. Find the model number and then go to this part of the AVSForum: http://www.avsforum.com/f/166/lcd-flat-panel-displays
Look up the "owners only" thread for your TV. Usually someone has posted their settings, which will make a good starting point.
Finally, a hum is a possible sign of a power supply in distress. It usually means the 60Hz alternating current signal is getting to the audio amplifier. However, it can also be normal for the cheap type of audio amps used in almost every flat panel TV available now.
Did you have a warranty on the repair? That would be one avenue to pursue with Sharp or the repair company. We use Sharp TVs for the flat panels and haven't had any power supply issues (knock on wood).
However, I think your LG can give you as good a picture as your Sharp did if you apply the appropriate settings. Unfortunately LCDs are not a perfect technology. Today's are much better than those of 15 years ago but they still aren't perfect. Each manufacturer still has some issues and LG's seems to be those artifacts (you're not the first). Sharp's issues are usually "clouds" that appear during really dark scenes. Luckily there are way to mitigate the LG issues (unlike the Sharp's cloudiness patterns).