Would the temperatures during such a collision be able to ignite nuclear fusion, bringing the dead star back to life? If so, would it only be able to fuse for a short time before running out of fuel, or would it turn into a full-fledged star?
The answer to your question is both yes and no, depending on the circumstances.
Two white dwarfs colliding would likely yield a Type Ia supernova, assuming the combined mass exceeded the Chandrasekhar limit ($\sim1.4$ solar masses). The unstable object resulting from the collision could not be supported by electron degeneracy pressure; when the temperature rises rapidly due to the collision, there is nothing to stop it (compare this to thermal pressure in a "normal" star, which can make the star expand or contract to compensate for changes in temperature).
The rise in temperature triggers fusion, which then increases the temperature, which triggers more fusion . . . and so on and so on, in a very short time span. This is the same process, more or less, as in an accretion-powered type Ia supernova. The resulting explosion destroys the object, ejecting matter into space.
As with normal Type Ia supernovae, this sort of collision would likely happen in a binary system, with two white dwarfs in close orbits losing energy to gravitational waves and spiraling in (the odds of two unrelated white dwarfs colliding is very, very low). I'm not sure how luminous this event would be in the gravitational wave spectrum; likely fainter than a neutron star-neutron star inspiral but still strong. The speeds of the components could be quite fast, meaning that a lot of energy would be released in the collision.
Things are slightly more complicated if the combined mass is less than the Chandrasekhar limit. An example system that is predicted to undergo a collision is SDSS J010657.39-100003.3. The total mass of the two white dwarfs is $\sim0.6$ solar masses, certainly below the limit. In roughly 37 million years (Kilic et al. (2011)), they will collide and form a subdwarf - fusing helium, interestingly enough (compare this to the carbon/oxygen fusion in Type Ia supernova, which in those circumstances is unstable). Similar scenarios could play out in other systems.
This sort of a star would not be a "normal" star in the sense that it wouldn't truly be on the main sequence - just a bit below it. It would be an O- or B- type subdwarf, less luminous than main sequence stars of the same spectral type. I saw a couple of articles on these objects claiming that they would be stars just like the Sun - it's not true! Given that white dwarfs are highly depleted in hydrogen - remember, they are the degenerate cores of old stars - there can be virtually no hydrogen burning, even in a shell. Core helium burning is the only option.
On a final note, one explanation for the formation of R Coronae Borealis variables is the collision of two white dwarfs. I don't think we have solid evidence yet for preferring this over other models (e.g. something involving hydrogen loss from the outermost shell), but if it is the correct pathway, then the answer to your question could be a definitive yes. Note that this would require intermediate-mass white dwarf progenitors.