The difference between Trailer and Movie when creating a new one in iMovie

iMovie only supports the H.264 format, and will only load videos from two places:

• Camera Roll
• Synced iPhoto Library via iTunes (Photos Tab → Sync Photos from iPhoto → Include Videos)

As you've noticed, videos in your iTunes → Movies folder will only sync to the Videos app, and will not appear in iMovie.

The process is not at all convenient, but here's a video detailing how to convert a video to the right format on a Mac, and then get it into iMovie on an iPad: https://www.youtube.com/watch?v=oF8-sDmD_1Y. He uses an app called StreamClip Video Converter to convert to H.264. You might be able to get away with exporting that format from iMovie on your Mac instead. iMovie is picky about the format though. If you have trouble, try to reproduce the settings in the video exactly.

Yes, you can do this. I'd recommend using the storyboard shot list to get as much of the trailer done as you want following the recommended settings/lengths.

Then, when you are ready to take more control, use the File->Convert to Project menu option. This will transition the Project settings to the normal iMovie project version and you can change clip durations using the "Clip Adjustments" popup. You can also drag more event footage into the project/change or add transitions, etc. just as you would a normal project.