I would suggest keeping the Adobe Download Assistant but if you have separate icons for the editor and organizer apps you could trash the one that launches the welcome screen.
If for any reason you wish to see the welcome screen simply click on the top menu:
Help >> Welcome Screen
First of all, the icons on the desktop mean you have never properly ejected the installers for those programs, so right click them and choose Eject. You'll probably find that unless you're actually running PSE the relevant icons go away.
I would trash the download assistant, and not only from the dock, once you've got PSE installed, since there are always new versions and you'll probably have to download it again anyway if you need it later on. It's totally up to you whether to have icons for the editor, the organizer, and/or the welcome screen. If there aren't any more icons after you've ejected the installers, go to the applications folder and launch the component you want in the dock (if any), and then click-hold its icon and choose Options>Keep in Dock.
Thanks for the help.
I will try the suggestions. I also think i will remove it all and reinstall the whole thing.