As far as the audio lagging a little behind the video, that is very common when viewing the footage using an editing program. The program I used to use (which is no longer made) also had that lag, but when you export, or "print", your final edit to tape, it syncs everything up very well.
In Premier, when you want to capture a still, you have to isolate which frame you want to capture, put it in your timeline, then export that frame as an image file. This option can be found in under the "FILE" menu, then "Export".
Fair warning: If your camcorder does not record in a mode called "progressive scan", you will be capturing an image that is 320x240 blown up to 640x480. And even if it is a progressive scan camcorder (which very very few of them are), you'll get an image that is a full 640x480. That's 0.3 Megapixels. Very poor quality unless your main objective is to post it on the internet, in which case it's perfect.