Will the diode likely be destroyed with just DC current from two 1,5V batteries and no driver?
Bare laser diodes are fairly sensitive to over-voltage and over-current conditons. They can also be quite sensitive to ESD.
Two series AA cells (for example) might have enough internal resistance to keep from burning out a laser like yours (fairly high power). Two D cells might not. YMMV.
Is the point of the driver to only provide a steady current? Could not a suitable resistor do the same trick?
A good driver will also ensure there's no current or voltage spikes on power up and shut-down, and protect against ESD.
What is the point of the third pin on the laser diode?
Many laser diodes are packaged together with a photodiode to monitor the light output (usually from the "back facet" of the laser). This allows hooking up the laser driver in a control loop to achieve constant power output, even if the laser temperature changes, for example.
Typically the 3rd pin is a connection to the photodiode. Usually the laser and photodiode share a common pin (either "common anode" or "common cathode") and the laser datasheet will tell you which way your device is wired.
Is the light supposed to be spread out like that from a laser diode? I also bought a bag of cheapo 5mw laser diodes, that are mounted in a small brass housing and appear to have a simple driver attached. Would this housing be enough to focus the beam?
Bare laser diodes tend to emit light in an cone of 30 degrees or more half angle. This angle can be very different measured "in-plane" and "perpendicular" to the top surface of the laser chip.
Some laser diodes are packaged with lenses to collimate the beam so that it has less spread.
The datasheet for your device will tell you the expected emission angle.
Finally, in the case that I do need a driver to set this up correctly, would something like this do the trick?
The driver you linked looks well matched in terms of current and voltage capability.
Whether it provides adequate ESD protection, soft-start, etc., is not clear from the E-Bay listing.
Final Note: 380 mW is more than enough laser power to cause permanent eye damage in case of an accident. Be sure you understand the risks and choose appropriate safety precautions before working with this device and work safely.