I find that most spell checkers, including FFs, mistakenly takes many words as misspelled, that were actually correctly spelled to begin with.
I'm glad you brought that up. It illustrates the problem of creating a site-specific applet for this purpose.
To create this applet, we (developer or company, w/e) would have to either find an established data set that specifies the acceptable words AND a dataset that contains the grammatical rules, or we would have to create our own.
Using an established dataset, like the one MS uses in their Office products or an online one like Websters/Oxford, would simplify the creation, but only if they actually allow access to the dataset. If not, imagine recreating individually every acceptable word AND grammar rule! That tasks alone would take an individual developer half a lifetime to create.
I'm going to do some research into this, I'd love to give everyone some definitive answers about the availability of dictionary databases and grammar rule datasets.
UPDATE: I did confirm that Oxford and Websters do not allow access to their datasets without hefty licensing fees. However, there is an open source dictionary library called ASpell which is used in my favorite multi-protocol chat client Pidgin. ASpell has dictionaries for tons of languages, but there are no grammar rules.
So there you have it. Creating a spell checker based on open source software is doable, but the grammar checker would still be incredibly difficult to manage without plenty of dedicated time.