Is high temperature **required** for nuclear fusion? Tl;dr: Can fusion be achieved only by speeding up particles to high enough speeds to smash into each other?
I know that it is necessary for protons, or any oppositely-charged particles for that matter, to have very, very high amounts of energy to be able to get close enough to fuse or "touch". Usually, this involves raising the temperature of a reactor to millions of Kelvin and then accelerating those particles into each other. However, at such a small scale, are temperature and kinetic energy not the same thing? After all, temperature is just average kinetic energy. So, in theory, would it be possible to fuse two protons or particles if they were accelerated fast enough at each other, without raising the temperature very high? Or is there something I am missing here? I've looked all over the internet and can't find anything explaining why exactly this is possible or not possible, other than articles about cold fusion and how it was faked.
 A: The "easy" way to get a bunch of particles moving very fast is to make them very hot. If they are hot enough, some of them will fuse when they collide. 
While it is possible to speed the particles up in an accelerator/collider instead, and then smack them into each other, this is a hugely inefficient enterprise. The energy release upon fusion is tiny compared to the energy expenditure to rev the particles up to speed in a particle collider.  
A: One can achieve fusion without high temperatures. However, one cannot achieve net energy production using fusion without high temperatures (based on current knowledge).
You consider fusion using accelerated ions. It is possible, but, as niels nielsen wrote, inefficient: in two colliding beams, most ions will undergo Coulomb scattering, not fusion.
Another low-temperature approach providing fusion is muon-catalyzed fusion, but again, net energy production is not practicable without some future breakthroughs.
A: 
Tl;dr: Can fusion be achieved only by speeding up particles to high
  enough speeds to smash into each other?

Yes, there is a fundamental limit that needs to be crossed, the coulomb barrier. That requires a certain amount of energy, and the traditional solution is to heat it up until the average velocity is high enough to cross this barrier at a reasonable rate. That has proven difficult...
There is another approach called "colliding beam fusion". In fact, this is the first way fusion was ever generated, by accelerating deuterium ions into a metal foil infused with deuterium. This work also demonstrated the existence of tritium for the first time.
The problem with this approach is that in the vast majority of cases, the ions will reflect off each other at an angle rather than collide. Some very simple math demonstrates that the number of misses times the energy you put into the ions is always many orders of magnitude higher than the energy given off by the fusion events by those ions that do actually collide.
This has led to a cottage industry of methods that aim to "recycle" the ions while keeping their original energy so they get many chances to collide and thus the ratio of energy in to out is improved. One of the more easy-to-understand approaches is the migma concept, which uses a clever magnetic bottle to recirculate the ions through the center of the chamber. Other similar concepts include the fusor and the polywell. A more recent attempt is TAE's design. All of these have fundamental limits that appear to suggest they cannot work, although they are more complex than the "they just miss" of the basic colliding beam concept.
A: The prerequisite for a fusion reaction is that the two nuclei be brought close enough together for a long enough time for quantum tunneling to allow the potential barrier to be crossed.
The best way to achieve that goal at low temperatures is to use the environment for lowering the barrier height by using the same approach as for superconductivity.
The Cooper electron pairs use the lattice environment for creating a lattice vibration that generates a positive charge wave for reducing the Coulomb repulsion between electrons and allow them to be weakly coupled at low temperatures.
In case of a LENR process, we need to have negatively charged highly condensed clusters for reducing the Coulomb repulsion between nuclei and allow them to be strongly coupled at higher temperatures.
The best way to get such clusters would be to create stable structures made of only electrons.
But, as electrons repel each other, it seems impossible at first sight, except if one introduces the negative mass concept.
Assuming the Einstein Equivalence Principle (inertial and gravitational masses are equal) holds for negative masses:

*

*a positive mass attracts all masses, either positive or negative.

*a negative mass repels all masses, either positive or negative.

*a negative mass behaves like a positive mass for an inertial force such as the gravitational force (a negative mass falls towards the Earth)

*a negative mass behaves oppositively to a positive mass for an non inertial force such as the Lorentz electromagnetic force (a negative mass accelerates in the opposite direction to the applied force)

So in a pair (+e,-m)---(-e,+m), the electron (-e,+m) is attracted by the "positron" (+e,-m) and repels it so both particles do no longer annihilate by producing gamma-rays but move in the same direction with the same speed.
When interacting with a gravitational field or an external electromagnetic field, a "positron" (+e,-m) cannot be distinguished from a classical electron (-e,+m), so such a pair could be used to transport a superconducting current or for lowering the Coulomb barrier between nuclei in a LENR process.
Moreover, one can also have (+e,+m)---(-e,-m) pairs consisting of a classical positron and an "electron" of negative mass.
A (-e,+m)---(+e,-m) pair and (+e,+m)---(-e,-m) pair can interact each other to form a moving chain:
(-e,+m)---(+e,-m)---(+e,+m)---(-e,-m)
or a square-shaped (rotating) structure:
(-e,-m)---(+e,+m)
(-e,+m)---(+e,-m)
Two square-shaped structures can form a cube-shaped (rotating) structure and cube-shaped structures can organize to form more complex structures such as filaments and rings through polymerization process.
As the quantum vacuum can be considered as an open and nonlinear system far from equilibrium state, one could envision to follow the Ilya Prigogine's approach to make emerge order from chaos and allow these self-organized stable structures exhibiting collective coherent behavior, to appear from the quantum vacuum by stressing it with shock waves produced, for example, by high voltage discharges, to amplify the vacuum quantum fluctuations.
The energy spectrum associated with the Dirac equation forbides energies between -mc^2 and +mc^2 defining a energy gap separating a valence band from a conduction band
but a stable pair of net mass null, consisting in an electron of positive mass (occupying the +mc^2 energy level) and a "positron" of negative mass (occupying the -m*c^2 energy level in the Dirac Sea) is not forbidden.
By equaling the (-mc^2, +mc^2) energy gap width to the electrostatic energy one gets 2mc^2 = K*e^2/d with K = 9E+9 MKSA
and we can find that the separation between the "positron" and the electron is d = Ke^2/(2mc^2) ~ 2.8E-15 m which is to be compared with the (reduced) Compton wavelength for a
classical electron-positron pair which is lambda = 2Dx = hbar/(m*c) ~ 3.8E-13 m.
So the distance between the "positron" and the electron is about one hundred times smaller than the one in a classical electron-positron pair.
