Does anyone know if schools will let you submit an optional essay after submitting the application?

I know this will sound odd, but I thought of a brilliant diversity essay that I could write. (I was so busy thinking of diversity in the form that law schools seem to emphasize most—ethnic/racial—that it didn't even occur to me to write about a significant diversity factor in my own life.

Thanks!

I read in a book about getting into law school that one should always write optional essays. Is this true? I mean, some of the schools I'm applying to I am interested in because they're a great school but I'm not any more or less interested in them than another great school. So, should I make up stuff and write an optional essay about why I want to go to ____? Anything I write would just be stuff I pulled from their viewbook/website. Or, should I skip it?

What are everyone's opnions on submitting paper applications vs. LSACD (electronic apps)? Obviously I will follow the school's stated preference if they have one. But, what should I do otherwise? For example, I'm working on my application for The University of Michigan and their viewbook seems to emphasize that LSACD applications can take up to two weeks to be received. Would I do myself a favor by sending it via FedEx? Would that even matter since my LOR are being sent through LSAC?