Do gamma ray bursts play a role in cosmic evolution? If gamma ray bursts were to interact with gas clouds in the early universe, might it be a relevant factor in star production?
 A: There have been plenty of studies on the connection of long-duration GRB rates and star formation (e.g., Robertson & Ellis 2011, Trenti, Perna & Tacchella 2013 and Wang 2014, all arXiv links). The relation comes from observations of star formation history, $\dot{\rho}_*(z)$, and the number of gamma-ray bursts, $dN/dz$, with $z$ being redshift in both cases; the relationship between the two takes the form
$$
\frac{dN}{dz}\propto \left(1+z\right)^\delta\dot{\rho}_*(z)
$$
where $\delta\gtrsim1$. So it seems that there is a connection between the GRB rates and star formation.
However, that is (sorta) the converse of what you want. A 1999 paper, Efremov, Ehlerova and Palous (also an arXiv link), discuss the role of GRB energy input in triggering star formation, with regards to the LMC. Their conclusion was that,

... the abrupt energy input connected to a GRB or a continuous energy input connected to an OB association can trigger star formation. Shells related to GRBs need the higher energy to fragment into protoclusters than shells connected to OB associations, where the lower energy is sufficient. The star formation triggered by GRBs can initiate or complement the star formation triggered by OB associations.

Which seems to answer your question in the affirmative, GRBs do play a role in galactic evolution by triggering star formation.
A: 
Do gamma ray bursts play a role in cosmic evolution?

I don't really know the answer to that. But I imagine black hole jets in general play some kind of role. Have you ever read any articles like Black Hole Creates a Galaxy? 
"The astronomers think the black hole is powering star formation in the nearby galaxy by spraying its jets of high-energy particles toward it. In fact, the quasar could have triggered the galaxy's formation in the first place when its energetic jets hit nearby clouds of gas. " 
There's other articles along the same lines:
"Earlier observations had shown that the companion galaxy is, in fact, under fire: the quasar is spewing a jet of highly energetic particles towards its companion, accompanied by a stream of fast-moving gas. The injection of matter and energy into the galaxy indicates that the quasar itself might be inducing the formation of stars and thereby creating its own host galaxy; in such a scenario, galaxies would have evolved from clouds of gas hit by the energetic jets emerging from quasars".
And then there's articles with a different slant:
"Blasting up to 2,000 light-years from the black hole, a powerful radio jet penetrates through the galaxy’s interstellar medium. Within this jet, up to 1,000 light-years from the black hole, there is a dramatic paucity of young stars; the jet appears to have cleared a cavity near the black hole, removing star-forming gases, snuffing out star birth".
For all I know, we may owe our very existence to a gamma ray burst. And for all I know, a gamma ray burst could kill us all. Check out looking down the barrel of a gamma ray burst. 
