You can store flash items in Livelink - they are stored as normal documents, you may need to amend some of the config files to get the handling that you want for VIEW and DOWNLOAD, which it looks like you have done.
If you want to display a flash item on the screen when a user enters a particular folder/project etc then you are probably best loading the item into Livelink and then creating a Customview.HTML file containing the link to the item.