I'm on Windows 7, and I recently downloaded MGE and put it in my Morrowind directory, as I'm supposed to. I try running it, but I get a message "MGEgui has stopped responding..."

I then tried to play Morrowind without it but then I got this message: "Morrowind Launcher.exe - System Error: The program can't start because d3dx9_40.dll is missing from your computer. Try reinstalling to fix this problem."

I want to know what I did wrong, if reinstalling the game will delete my save files/mods, and how to avoid this problem and similar ones in the future. If somebody can help me, I would appreciate it very much.

Install the D3D9 runtimes:

http://www.microsoft.com/downloads/en/details.aspx?FamilyID=3b170b25-abab-4bc3-ae91-50ceb6d8fa8d

7 doesn't include it, but you need it for Morrowind (with MGE especially).

Where to install,and how ?

Just download that link, run it, and it'll create an installer (the link goes to an archive of the files that Microsoft provides). Where you extract them to, the actual installer will be there (DXSETUP.exe), run that and it'll do the rest automatically.