Are neutrinos' velocities related to their energies? As they are with other particles? I have read extensively about neutrino energies, including in the popular press, and yet velocities of the neutrinos themselves (not the detectors they 'run into' or the leptons they 'create' after interaction) are rarely mentioned....
How can a neutrino's speed be measured to begin with?
How sure are experimenters and theorists about their calculations of neutrino energies, anyway?
 A: 
How can a neutrino's speed be measured to begin with?

is  answered in the link given in a comment to the question ,

Astronomical searches investigate whether light and neutrinos emitted simultaneously from a distant source are arriving simultaneously on Earth. Terrestrial searches include time of flight measurements using synchronized clocks, and direct comparison of neutrino speed with the speed of other particles.
It used to be that the neutrinos were mass-less, the measurements are able to establish they have a mass, but the accuracy of measurement allows only to give limits.


From cosmological measurements, it has been calculated that the sum of the three neutrino masses must be less than one-millionth that of the electron,

Because their masses are so small the  velocity of the neutrino detected in cosmic ray events is very close to the speed of light. Their energy   as with all particles with relativistic speeds, is given with the four vector relation of special relativity.

How sure are experimenters and theorists about their calculations of neutrino energies, anyway?

The experiments measure the energy of incoming cosmic neutrinos by their interaction with the matter of the detector, measuring the energy deposited  and using programs to fit the spectrum , knowing how particles interact with matter.
In the laboratory and in producing neutrino/antineutrino beams the interactions that generated the neutrinos are chosen so that the energy spectrum of the beam is  known. Individual interactions can give the neutrino energy as a missing energy from summing for energy conservation in the interaction, depending on the experiment, example:

To create the neutrino beam, a beam of protons from the Super Proton Synchrotron at CERN was directed onto a graphite target. The collisions created particles called pions and kaons, which were fed into a system of two magnetic lenses that focused the particles into a parallel beam in the direction of Gran Sasso. The pions and kaons then decayed into muons and muon neutrinos in a 1-kilometre tunnel. At the end of the tunnel, a block of graphite and metal 18 metres thick absorbed protons as well as pions and kaons that did not decay. Muons were stopped by the rock beyond, but the muon neutrinos remained to streak through the rock on their journey to Italy.:

From knowing the four vectors of the charged particles whose decays generate the neutrinos computer programing can give the statistical distributions of the neutrino beams to be used in the experiments.
A: Yes, the speed of neutrinos is related to their energy via the usual formula for relativistic energy:
$E_{\mathrm{kinetic}}=E-E_0=\gamma m_0 c^2 - m_0 c^2$
Now, neutrinos have masses of electronvolts ($eV$) or less - we don't quite know yet. But their (kinetic) energies usually are many orders of magnitude more than that: radioactive decay or processes in stars gives energies in the 100keV-MeV ballpark, and at accelerators even GeV are possible. So the restmass of the neutrinos is utterly negligible, and one does not need to distinguish between relativistic, total, or kinetic energy. Their speed is thus simply the speed of light. Or, more precisely, for a 1MeV neutrino e.g. from the Sun, the $\gamma$-factor is $10^6$:
$E_{\mathrm{kinetic}} / (m_0 c^2) = \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
Your pocket calculator can't even easily calculate that as this speed is so close to the speed of light, so use the relation
$\frac{v}{c}=\beta=\sqrt{1-\frac{1}{\gamma^2}}\approx 1-\frac{1}{2\gamma^2}=1-5\times10^{13}$
to see that this neutrino travels at the speed of light to excellent precision.
