Electronic – BJT (Common-Collector Configuration)

If you drive the base positive enough (assuming an NPN) what happens is the transistor 'saturates', and the collector voltage will dip to just a couple tenths above the emitter. In this mode, the base-collector junction is actually forward biased, instead of reverse biased as is usually the case.

Voltage is indeed taken with reference to something. It is a potential difference between two points. So you cannot just say "this node is at 5 volts", it has to be "this node is at 5 volts relative to this node".
Usually, to make things easier we define a circuit reference point, e.g. circuit ground so we can measure all voltages relative to this. Then in this case you will hear "this node is at 5 volts" and can assume it is relative to the reference point.

So in your first circuit above, you need a reference point to determine the voltage at the node. If your professor said "the negative of the battery has zero voltage", this would be incorrect. You could say that the negative of the battery is your zero volt reference, or circuit ground, in which case the node pointed to is at Vcc.

Usually you will have a symbol representing circuit ground, such as this one:

We can see it used at the bottom of this circuit, and the voltages in reference to it :

There are different ground symbols, depending on what the actual reference point is (e.g. the earth underfoot, a metal chassis, or a local circuit ground on a PCB):

Your last example is drawn confusingly (as Phil says, you rarely have two batteries in a circuit like this) It's actually a common collector circuit, and the common point is at 5V here (relative to the circuit ground symbol):