on mine, firefox puts a red line under my misspelled words and then I right click on the words and it gives me alternate possibilities. If you right click in the text box there should be spell check in the options to turn it on. Hope this makes sense
I love the spellchecker and all the customizations you can use to tweak Firefox. It's just one of the things that makes me love it. You can also add words to your spellcheck dictionary by right-clicking on anything it sees misspelled and selecting Add to dictionary from the menu. It's handy so at least when you're proofreading a post or something, you don't have to remember which red underlined words are just words not in the dictionary that you know are spelled right (such as an unusual brand name or a non-traditional spelling of a person's name) or whether it's something you have to correct.
And if you really want to be sure that your codes are compatible with IE too, there is a firefox add-on that simulates IE in a different tab (I haven't tested it myself, but it's a nice option to have!) I can't say I'm impressed with the other IE alternatives out there, though. Opera is still not completely there. Neither is Chrome or Safari (although Safari is FAST).
Are you only experiencing this with certain file extensions that try to launch media player (mp3, wma, acc?) Also, where are you trying to open that file from (gmail, hotmail, yahoomail?) Are these files embedded into the email?