Does anyone know if the new Kindle Fire will run E-Pub files like my Nook?

Marie
There's no ePub support
But it's an Android device. There's no reason that an ePub reader couldn't be installed on it.
Of course it will. Install aldiko reader, or any of 10 other android readers
If they allow you to install apps from android marketplace? I thought it was a modified version of Android, I suppose if you root the device, but natively it won't have any ePub readers on there.
Right now the Amazon App store has ePub readers listed.

http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dmobile-apps&field-keywords=ePub&x=0&y=0

including aldiko and moon+ reader

Hopefully they won't block those from the Fire somehow.
We don't know yet whether Amazon will lock out certain apps. I sure hope they allow us to install reader apps from other stores and 3rd party apps, but I'm not going to hold my breath.
There you go I guess it will be possible.
They're not going to lock anything in their own app store out. Doing so would give the impression of crippling the device...which would cripple sales. You'll be able to read epubs if you want to. Though, with Amazon's sync-between-devices feature, I'm not sure why anyone would want to. It's very nice to read on your Fire, then go shopping with the wife and read at the same place on your phone.
There's a quote here on MR somewhere, where someone from Amazon said they will allow you to load APKs on the device, so you should be able to install Android Market and grab whatever Apps you want.