Are heavy or light nuclei used in the direct detection of dark matter? My understanding is that recoils of light nuclei are easier to detect as they recoil with greater energy, however is the interaction cross section not much smaller, meaning interactions are much less likely to occur? If this is the case, how is this trade-off managed? Are heavy or light nuclei preferred, or is it a middle ground?
 A: Different target nuclei are read out with different technologies, and taken together, this has advantages or disadvantages depending on the unknown mass or coupling of the dark matter.
This question is about the so-called direct detection of dark matter. For the sake of this answer, let us thus assume that dark matter is made of WIMPs or similar particles with masses anywhere in the range typical of massive stuff we see around us, i.e. anything from MeV to TeV or thereabout. Such WIMPs are expected to scatter of nuclei, and direct detection experiments attempt to detect such recoils.

My understanding is that recoil of light nuclei [...]
recoil with greater energy

Correct. For a given WIMP mass, speed, and scattering angle, a lighter target nucleus (e.g. germanium) will be given a higher recoil energy than a more massive target nucleus (e.g. xenon). However, all that matters in the scattering kinematics (think billiard balls) is the reduced mass of the WIMP-nucleus system. Therefore, experiments that search for sub-GeV dark matter prefer lighter target nuclei. Specifically, the SuperCDMS experiment uses germanium and silicon, CRESST uses the calcium and oxygen in CaWO$_4$ crystals, and prototype experiments use sapphire Al$_2$O$_3$ or helium. SENSEI even is optimized for electrons as a target, using silicon CCDs. In contrast, the best sensitivity at above-GeV masses come from experiments that use xenon, a massive nucleus.

My understanding is that recoil of light nuclei is easier to detect

That is not correct in general. For example, DarkSide that uses the lighter argon as a target material has a higher energy threshold than XENON or LZ that use xenon as target material. So that "ease" just depends on the particulars of the experiment design and readout strategy, and can not be stated in such a general way.

however is the interaction cross section not much smaller,
meaning interactions are much less likely to occur?

Indeed, the deBroglie or Compton wavelength of the momentum that is transferred in the WIMP-nucleus interaction can be much larger than the size of the nucleus. Therefore, the WIMP interacts coherently with the entire target nucleus. For a target nucleus with mass number $A$, the cross section thus gets an enhancement by a factor $A^2$. Thus, the cross section is much larger for e.g. xenon compared to e.g. silicon.

If this is the case, how is this trade-off managed?
Are heavy or light nuclei preferred, or is it a middle ground?

This trade off is one reason, and a very good one at that, for why we have a number of different experiments to search for WIMPs in the MeV-TeV mass range. It is not clear a priori which one may win out and discover dark matter. For a given mass range, it is often clear which technology is the one of choice, but given the range of possible WIMP masses, this becomes a gamble.
Further, in the case of a detection of a signal in one of these detectors, it is important to have multiple experiments that can independently and with different systematics and different kinematics check for a possible dark matter origin of such a detection. This is known as the multi-target approach to the direct detection of dark matter.
To design an experiment, other aspects come into play. Most importantly, what is the level of background radiation? Low-radioactivity targets will have a huge advantage. What is the cost of the material? Liquid xenon can be cheaper than highly instrumented crystals or isotopically depleted argon. How robust is the technology, can it be scaled up? Etc. It is an entire very active research field of experimental particle physics that works out exactly this optimization.
A: 
The primary evidence for dark matter comes from calculations showing that many galaxies would fly apart, that they would not have formed, or that they would not move as they do if they did not contain a large amount of unseen matter. Other lines of evidence include observations in gravitational lensing and the cosmic microwave background, along with astronomical observations of the observable universe's current structure, the formation and evolution of galaxies, mass location during galactic collisions,[ and the motion of galaxies within galaxy clusters.

Dark matter   was discovered in cosmological frames, and the discovery has nothing to do with nuclei
There are particle physics experiments searching for particles that could be responsible for the macroscopic observation of dark matter,
For example
Search for dark matter produced in association with a Higgs boson decaying to a pair of W bosons at CMS

The results of a search for dark matter produced in association with a Higgs boson, with the focus on Higgs boson decays to two W bosons and each W decaying leptonically, are presented.

Searches for dark matter with the ATLAS detector

The presence of a non-baryonic Dark Matter (DM) component in the Universe is inferred from the observation of its gravitational interaction. If Dark Matter interacts weakly with the Standard Model (SM) it could be produced at the LHC. The ATLAS experiment has developed a broad search program for DM candidates,

The experiments are proton proton collisions at the LHC and have nothing to do with nuclei light or heavy.
