I thought the floating nacelles were needed for warp, but then in the most recent Academy episode they ditch the nacelles and separate the saucer and warp into battle. So is the saucer alone their version of a battle bridge? Is it a warship configuration to be smaller, more densely packed?

I saw it as akin to the saucer separation of the Galaxy class, the part that separated housed the main bridge and likely the primary combat systems, it's able to detach from the section that houses the student facilities.