Blu ray media can only be burned with a Blu Ray burner, so no, the BD-Rom drive won't do that, and no plain dvd writer can burn it either. If you anticipate the need for burning Blu Ray media, you'll have to get a Blu Ray burner.
AnyDVD has two different tiers these days. The regular AnyDVD and AnyDVD HD. Only the HD version can decrypt and rip Blu Ray. If you already have the HD version, you are set. If not, you can upgrade.
DVDFab has an add on for Blu Ray. The Blu Ray to Blu Ray add on looks like it costs â‚¬105.60 for the free lifetime updates.
Here is the page where you can download BD Rebuilder: http://forum.doom9.org/showthread.php?t=143716
BD Rebuilder requires that you have AviSynth, FFDshow, and Matroska Splitter installed first, but they are all free, and there are direct links to them in the first post of that thread. Make sure to read that first post for instructions on installing FFDshow.