Chad: you are a bit confused (because it is confusing and the manual is unhelpful).
1. 'PRIVATE' is a folder, not a file. The folder contains a folder structure (then next folder down is AVCHD, the next is BDMV, and the last is a folder called 'STREAM'. In there you will see all the actual video clips (*.mts files). This folder structure is according to AVCHD specifications. Same as used on Blu Rays more or less.
2. You are correct, you can just insert the SD card in your computer and copy the files in the STREAM folder I indicated above anywhere you want on your computer's hard drive. You can then play them or edit them or whatever. You do not need any special software.
3. The PJ760 has a built-in normal USB cable. You can just plug it into your computer. You do not need to buy any other cable. So you can also download the files using the camcorder as an SD card reader if you like.
I, like spyker1212, prefer to use the Sony software to organize the video files by date, but you certainly don't have to. Everything is easy, as long as your computer is not a Mac.