Terrorists gain complete control of the world's nuclear weapons and they want to destroy the moon. Could they do this with ICBMs?
No. They do not have the delta-vee to reach the Moon (or even achieve low earth orbit).
If not, is there enough power in the arsenal to destroy the moon if they could somehow transport the weapons to the moon's surface?
Nope, again. The current Earth arsenal falls short by many orders of magnitude. Even magicking the warheads on the Moon and using them as Orion propulsors is not going to significantly affect the Moon at all.
Most of the adverse effects nuclear warheads have on Earth are due to indirect effects: fires, nuclear winter, radioactive contamination. This is of no matter as far as the Moon's destruction is concerned.
If this is totally impossible, could a weapon strong enough to do it be created with the current level of technology of humans?
We need energy - lots of energy. We do not have that much energy. We need to take that energy wherever it is, and on Earth there's not enough energy in fossil fuel form, while radioactive ores cannot currently be mined to that extent (you'd need to tap the Earth's core).
So we're left with the Sun and with the gravitational potential energy of solar bodies. Solar sails, and kinetic impactors.
We could make a significant effort, which would have to last several centuries, and use railguns and solar sails to place a sizeable asteroid on a collision course with the Moon. This happens in Bob Shaw's The Ceres Solution.
The impact might fragment the moon (and ensure a meteorite shower to dwarf the Chicxulub Event), but it would require a body with a considerable fraction of the Moon's mass, and at that point it would begin to become more convenient to do the same operations on the Moon instead. It would be much easier to reach, and way safer to slowly pull the Moon towards Sun-Earth Lagrange 1 point.
There is at least one novel featuring a comet-triggered fragmentation, but again, the comet would need to travel at relativistic speeds (within a few thousandths of percent from c) and be unbelievably dense. Not even osmium is dense enough; at those speeds, a rough estimate of the penetrative power of a projectile is given by Newton's formula. Pump more energy in the impactor and all you get is a massive explosion without penetration (I was a bit taken aback by the last Starfire novel for this reason).
We could use some handwavium to decrease the Moon's density and have it made up of loose rubble, but keep in mind that this doesn't really work: with that mass, the rubble would collapse into a compact sphere under its own weight, and the lunar seismology (and density, and mascon distribution) is roughly known since the Apollo years.