Mic troubles

However, having said all that, the other problems you describe indicate a faulty mike.

Most shotgun mikes are only going to provide a mono output. In some cases there are two internal pick ups and the output would be stereo. A good example of this would be the Audio Technica AT822 which almost looks like two mikes joined together. I have a Sennheiser MKE300 and I modified the plug in order for the signal to be fed to both channels (double mono)
I'm not familiar with your Sony mike and I would think that it would feed both channels, either as stereo or double mono, but perhaps it is not designed to do so. And being a hot shoe connection it wouldn't be possible to alter it until the editing stage.
However, having said all that, the other problems you describe indicate a faulty mike. The low signal level indicates a lack of power to the mike, which in your case should be provided by the camera (unless there is a place for a battery in the mike, but I doubt it). I don't think it should be tinny either.
I have had some issues with my Sennheiser picking up too much camera noise and handling noise but the audio of the subject is quite good.

So, if you want a suggestion ...
Take the Sony mike back - if you can.
Buy a good quality shotgun (not the Sennheiser MKE300!) that has a standard plug on it (XLR or 3.5mm).
Buy an external recording device (I use a minidisc and/or an Edirol R-1)
This will give you greater control over mike placement and provide redundancy ( I recently recorded a wedding with a broken mike cable -I didn't find out until the editing stage that the camera hadn't recorded usable audio)