Basically, I'm seeing that intergalactic medium can sometimes have a temperature in excess of 10^5 K. Would the high temperatures have an effect on intergalactic travel?
No and Yes.
The inter-planetary medium is a large fraction of that hot and much denser, yet it is routinely ignored for thermal purpose because radiative heat transfer dominates the problem by many orders of magnitude.
On the other hand shielding against the ionizing effect of running into stray gas and dust at relativistic velocity is a huge concern just as it would be for interstellar travel.
But either way it will be a solved issue long before the matter comes up.