1) Can an insulating ball actually be attracted to a negatively charged plate? 2) Upon touching, will the insulating ball then be negatively charged?

Neutral objects can be attracted to charged objects
through a process called induction.  This works because all matter is composed of
positive protons and negative electrons.  While the protons are basically fixed in
place, the electrons are able to move somewhat within a substance.  Remember also that
like charges repel and opposite charges attract

In your
example when a negative charge is brought near a neutral object the negative electrons
in the neutral object will be repelled. They will try to move away from the approaching
negative charge. When they do this, the neutral object appears to have a small positive
charge on the side where the negative charge is approaching.  A temporary charge
separation has been "induced" in the neutral object, egen though its overall charge is
still neutral. If the negative charge is then removed, the electrons in the neutral
object will move back to their original  positions. Since opposite charges attract, the
neutral object will now move toward the negative
charge.

Now consider what will happen when the negative
charge is brought in contact with the neutral object.  As the negative charge
approaches, as explained above, there is a positive charge on that side of the neutral
object. When the two objects make actual contact, some of the electrons from the
negatively charged object will actually move to the neutral object to try to neutralize
the apparent positive charge. Now when the objects are separated both objects have a
negative charge.