Regarding (1) and (2), yes, it's very possible. Moreover, provided $N$ is hyperbolic, I believe Thurston proves the Gromov norm of $N_1$ is strictly smaller than $N$'s volume. The only time Dehn filling does not strictly lower the Gromov norm is when you're doing a filling on a non-Hyperbolic component (in the JSJ-decomposition) and the incompressible boundary tori of the hyperbolic components remain incompressible after the filling.
I give an example of of this in my "JSJ decompositions of knot and link complements in S^3" paper towards the end. Here it is:
In the picture you see a 3-component link with the components labelled. This link is hyperbolic, but if you delete component 3 (deletion of components is a Dehn filling operation) you get a 2-component link with five incompressible tori and two Whitehead links in the JSJ-decomposition (the other manifolds are Seifert fibred, they are two Trefoils and two manifolds I call "keychain links" which are trivial punctured disc bundles over the circle).
This example shows how you can make arbitrarily extreme examples of the kind you're seeking. You take a very complicated satellite knot or link with many incompressible tori, then you add another component to the link which punctures all the incompressible tori of the original knot/link.