MPEG and MPEG-1 the same?

MPEG could mean MPEG-1 OR MPEG-2. QuickTime can open (import) MPEG-1 files natively and needs a the QuickTime MPEG-2 component to open MPEG-2 files.

Another possibility that I've seen recently is that a file will have .mpg as an extension but is actually an .avi file. Changing the extension and attempting to open will show if that's the case.

You need to examine the file "contents" and not the file "extension" when trying to view .mpg files with QT Pro.
In the old days we would distinguish the two "flavors" of MPEG files by using .mpg for MPEG-1 formats and .mpeg for MPEG-2 formats.
It doesn't work that way any longer and many other file extensions can use MPEG-1 formats inside another "wrapper" (.mov or .avi).
If you get the error message (-2048) when you attempt to open a .mpg file it is safe to assume that the file is actually MPEG-2 format. These files require the MPEG-2 Playback Component ($20 more, from Apple).
If you get no error message (but the player window opens and doesn't show any content) use the Movie Info window to determine the codecs and formats used in the file.
History:
MPEG-1 is like VHS and not much software creates it any longer. Normal size is 352X240. It brought us the .mp3 audio file. Let's just let it die and go away.
MPEG-2 is the current format of most DVD's. Normal display size (in QT Pro) is 640X480. Looks and sounds great but needs improvement.
MPEG-4 is the new kid on the block. Up to High Definition display resolution and numerous audio formats that allow up to 24 channels of audio.
The file extensions used to be: .mpg, .mpeg and .mp4.
No more. Now you can have MPEG-1 formats inside .mov or .avi containers. You can also have .mp4 files inside .mov containers. .avi files don't seem to play by any "rules" any longer. They are a box of chocolates. You never know what you're going to get.
Hopes this helps, or at least makes you hungry.

QuickTime Player can be used to view your TiVo (MPEG-2) files but it requires the MPEG-2 Playback Component (a separate $20 purchase).
It is only good for "playback" and QT Pro still can not export the audio from the file.
Do a search for some Windows software that can "extract" the two parts of the TiVo file. Once extracted the parts can be "combined" into QT formats that QT Pro can edit.
You should also be able to view your TiVo files with your DVD player software if they are in .VOB or VIDEO_TS formats.
Sounds like the alphabet song.

It is necesary to demux mpeg-2 only if you want to export the file to another format complete with sound. In QT you would need to buy the mpeg2playbackcomponent. You also would have to find a utility that would convert the separated mpeg-2 sound file to a non-mpeg format as QT will not by itself convert mpeg-2 sound to other formats. As you can see it is a rather involved process.
If all you want to do is view mpeg-2 files (that is, you don't want or need to convert them to other formats) you may be able to view them with the Windows version of this video player. VLC

I'm getting lost in the string of emails. Is it possible to convert Tivo recordings from Tivo Desktop to the video Ipod using QuickTimePro?