Yes. Import the clips you want to work on into iMovie one at a time. Edit them and save to an mp4 file. Then when you are done editing, import the finished clips back into iMovie and join them together as you wish.
Another way you can do it, is to trim the clips in QuickTime Player and save them. You can not convert them though in QuickTime Player. You would have to save them in there original format, unless you have QuickTime Player Pro, which will allow you to save them in just about any format you want. Then again, import them into iMovie and join them together as you wish.
Can not help you with FCP, as I do not use the program at all.