Is there anyone I am supposed to inform of the Imperial Guard killed over by the Roxey Inn? Is there any way I can heal the horse?

There is no in-game reason to take any action. Guards are killed with some regularity by random encounters on their patrols. They will respawn after three days. If his horse survived, you can heal it, but even if you do, it will disappear and respawn when the new guard does. Further, mounting the horse will count as stealing.

Now, despite all this, my character roleplays pretty deeply. She will take the guard's shield for return to the nearest guard/soldier she finds. She will also heal the horse and, with her 2 minute command creature spell, have it follow her and her own horse to the nearest stables.

If you are on PC, then of course there is the resurrect thing but maybe that was not what you were thinking of