MEP may have problems importing AVIs/MPEGs. One workaround is to convert the file to a DV-AVI and then importing it. You can easily convert it using Virtual Dub using the free Panasonic DV codec.
Virtual Dub, a free program from Avery Lee, can be downloaded from here.
If you want to import VOBs/MPEGs into Virtual Dub, install the plugin from here.
Installation of Virtual Dub is not required: just unzip the package and double-click the exe file.
To install the Panasonic DV codec, download and run this exe file and follow the instructions in the folder you installed the files into (default C:\Program Files\Panasonic...).
To convert your video file to DV-AVI:
After it has been converted, you should be able to open (and burn) it in MEP with no sync problems.
One more thing: if you have converted a 16:9 widescreen video, you may have to set it to 16:9 format on the MEP timeline to make it display properly. Right click on the video, choose Properties, then choose 16:9 from the dropdown list.