Premiere Elements often has a problem opening older version projects with newer versions of the program.
There's definitely no way to open a newer version project with an OLDER version of the program.
Although I'm surprised to hear that version 9 runs slower than version 4. Adobe has made a real effort to tune up the performance of the program in recent versions (especially since version 10). But I guess how version 9 performans compared to version 4 depends a lot of what your hardware specs are and, even more so, they type of video you're editing.
With regard to your Premiere Elements 9 performance and related matters
1. Are you working from the Premiere Elements 9.0.1 Update? If not, see Help Menu/Update within the opened project or use the download from Adobe
2. Do you have the latest version of your video card driver according to the web site of the manufacturer of the video card? If so, then try deleting the BadDrivers.txt file in the following path (if you have Windows 7 or 8 64 bit)
Local Disc C
and, in the 9.0 Folder, you will find the BadDrivers.txt file that you delete. Remember to be working with Folder Option Show Hidden Files, Folders, and Drives active so that you can see the complete path.
3. With regard to this file that you would like to use in Premiere Elements 4...can we assume that you are talking about a Premiere Elements 9 project file (project.prel) and not a file that you have imported into that project? There is a ClipMate Copy/Paste between projects that can be used for versions 8, 9, 10, and (except for transitions) 11. It does not work in 7, but I do not know that I have ever tried to use it in version 4.
Just some thoughts for now.