Why should a neutrino being nearly massless mean that they travel near the speed of light? It seems to me that photons travel only at the speed of light due to some intrinsic property of photons but once a particle has mass, its mass (irrespective of how small this mass is) should have nothing to do with how fast it travels and indeed, a neutrino could in theory be caught maybe in some sort of Penning trap and be essentially stationary. I know that electrons have very little mass and can move very fast and also move very slowly.
If a particle has mass much less than that of an electron (as a neutrino does), is this some special case where it becomes somehow more "photon-like?"
One guess that occurs to me is that when neutrinos are produced it is due to very energetic processes so that at the time they come into existence they are already move near the speed of light and since they do not interact with normal matter very much, there is no way that they get slowed down. Furthermore, if they are detected, it is because they interact in a way that causes them to combine with another particle and so they no longer exist in a "free state" -- this implies maybe that they only exist with the speed they have at production time. But this does not mean that perhaps some way could be developed to either create them in a process wherein they have less initial velocity or to in fact slow them down and confine them.
 A: 
One guess that occurs to me is that when neutrinos are produced it is due to very energetic processes so that at the time they come into existence they are already moving near the speed of light and since they do not interact with normal matter very much, there is no way that they get slowed down.

This is essentially correct!
Often, neutrinos and antineutrinos are produced in nuclear processes such as $\beta$ decay, where the typical energies are of the order of the MeV, which (as @rob describes in his answer) means they have Lorentz factors of the order of a million or more.
The fact that we have so far only seen energetic neutrinos is in part due to the specifics of how they interact: their interaction cross-section increases linearly with energy, therefore low-energy neutrinos interact even less than the high-energy ones!
Nevertheless, low-energy neutrinos are thought to exist, for instance in the cosmic neutrino background.
A: The “relativistic limit” is when the kinetic energy is much larger than the rest mass.
The total relativistic energy of a particle with mass $m$ and speed $v$ is
\begin{align}
E &= \sqrt{p^2c^2 + m^2c^4} = \gamma mc^2
\end{align}
where $p = \gamma mv$ is the momentum and $\gamma = \left(1-v^2/c^2\right)^{-1/2}$ is called the Lorentz factor.  For low speeds, $v \ll c$, you can show that
$$
E_\text{slow} \approx mc^2 + \frac{p^2}{2m} + \cdots
$$
which gives you the “standard” kinetic energy.  But in the $v\to c$ limit the Lorentz factor gets big, the momentum $p \approx \gamma mc \gg mc$ becomes much larger than the mass, and the energy $E_\text{lightlike} \approx pc$ becomes independent of the mass.
An electron, with mass $\sim \frac12 \,\rm MeV$, is essentially already “lightlike” at kinetic energy $\rm 50\,MeV$, with $\gamma \approx 100$.  The heaviest neutrino might have a mass as large as a few eV (but probably not); such heavy neutrinos would become “lightlike” if they came into thermal equilibrium with the photosphere of a star.
A: It is not a law of of physics that neutrinos move very fast.
Stars emit neutrinos. Stars that move very fast to the left emit slowly moving neutrinos to the right.
Now we can see the reason why neutrinos move very fast: There are no stars that move very fast.
If there was a spaceship that moved very fast, stars would be seen moving very fast when looking out of the spaceship window, and the stars would be emitting slowly moving neutrinos to the opposite direction.
Now we can see another reason why neutrinos move very fast: There are no spaceships that move very fast.
