Is it moral or immoral for Satan to punish souls for immoral acts? Satan, to my understanding, punishes souls in hell as a result of their immoral acts on earth.  It also is my understanding that the concept of Satan represents pure immorality (evil).  

Does that mean it is immoral to punish souls in hell or is it actually moral?
In Dante's Inferno, that may have been the case, however, in the Bible, that is absolutely not the case.  Satan was never given dominion over hell.  In fact, Revelation 19 and 20 tells us Satan will be bound and thrown into the Abyss for 1000 years.  After his release, he will be cast into the Lake of Fire with Death, the Grave, the Beast and the False Prophet to burn forever.  Satan receives eternal punishment for his actions according to the Bible.  Aside from 14th Century literature, he never gets to take his revenge out on anyone.  In other words, Satan does not ever punish any souls for immoral acts according to the Bible.

As for the morality of "punishing" someone in Hell, I would contest that as well.  I would say a person who rejects God has chosen Hell over Heaven and God forcing a person who wanted nothing to do with Him in life to spend eternity with Him in a relationship they didn't want is tantamount to kidnapping.  Think Jayce Dugard.  That would be the truly immoral thing to do.