When you walk into Solitude for the first time, there is a Nord being executed for helping Jarl Stormcloak escape.
Is it possible to save this man from being executed?
Perhaps more importantly, is it useful to save him? (Or is he scripted to die?)
Yes and no.
Yes, you can save him, but its not useful in any way or form. He is very weak, and will die in 1 hit. As soon as you start attacking the guards, he will try to run away, causing other imperial guards in the city to take him down with 1 arrow.
If you do manage to kill the 3 guards fast enough, then you can interact with him, but there’s no dialog. All he says is “hmm”, “huh”, “what do you want (insert race)?” etc.
No, even if you killed the guards, the executioner, and the captain, and Roggvir survives, in a few seconds, he will die, even though no one attacked him.