If two photons collide, does the resulting particle have zero velocity? If two photons traveling in opposite directions along the same line collide, will the resulting particle have a velocity of zero relative to the rest of time space in the instant of the collision? 
 A: Generally no, because velocity is not a conserved quantity. It is momentum that is conserved in all interactions. For photons, the magnitude of momentum is simply
$$ p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda}, $$
so photons with different energies/frequencies/wavelengths will have different momenta. If the total momentum is nonzero before the collision, it will be nonzero after.
A: 
If two photons traveling in opposite directions along the same line collide will the resulting particle have a velocity of zero relative to the rest of time space in the instant of the collision? 

Photons are quantum mechanical particles. In the microscopic dimensions where quantum mechanical particles interact there are Nature's rules that dominate these dimensions, though they are usually insignificant in macroscopic dimensions.
One of these rules is the Heisenberg uncertainty principle, HUP,: one cannot define the location of a particle and the momentum of a particle with accuracy better than:

where $\hbar =6.62606957(29)×10^{−34}$ Joule second a very small number that is why it is effectively zero in macroscopic dimensions.
Thus two photons even with the same energy will not collide at a point.
Going into the mathematics of it, photon-photon interactions are very very weak, since there is no first-order interaction between two photons, but they have to go through a particle loop. In addition, momentum conservation requires two particles out.


A Feynman diagram (box diagram) for photon-photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

Feynman diagrams have one to one correspondence with calculable integrals that will give the probability for a given interaction.
A photon carries energy, two photons have an invariant mass.  In their center of mass system, depending on the energy available from each, the output can be again two photons, or if there exists energy enough to generate massive particles, there will exist a quantum mechanical probability for the interaction. They are proposing high energy photon colliders, gamma gamma colliders.
A: Note that the collision probability can be strongly enhanced in some nonlinear materials (such as a Kerr medium). As stated above, vacuum is a very weakly nonlinear material.
The resulting 'velocity' for the photon would be its momentum $\mathbf{k}$, the rule being that, if no loss occurs in the material, momentum and energy must be conserved: $\mathbf{k}_{1}+\mathbf{k}_{2} = \mathbf{k}'_{1}+\mathbf{k}'_{2}$. But these are vectorial quantities and $\mathbf{k}_{1}+\mathbf{k}_{2} = \mathbf{0}$ does not imply $\mathbf{k}'_{1} = \mathbf{0}$ and $\mathbf{k}'_{2} = \mathbf{0}$. Many solutions are possible, these are studied by nonlinear optics. Moreover, energy conservation puts further restrictions.
A: Photos are bosons. They can occupy the same quantum states and the same place in space, so they can't actually "collide". If two photons traveling in opposite directions along the same line collide, they just pass right through each other. 
I think you were probably thinking of pair production. It's a completely different process in which two energetic photons interact with nucleus and create a particle-antiparticle pair.
