Dan-El
The Emerald Archer
I seriously doubt he is a "spirit" in our usual sense of the term. I would say, since Krypton is an advanced society, the Jor-El we are privvy to is an advanced AI, stored in the crystals, which are like zip disks or floppies or CD-Rs or something. I know it's a horrible comparison, but it makes sense.
As for Lionel, he was possessed by Jor-El. That's how he got his powers. Information on the Kryptonian biological make-up could have been downloaded into Lionel's DNA when he had one of the elements (remember when it glowed like it did in Commencement, Season 4 finale?). That biological data may have been accessed when the Jor-El AI took over Lionel's vacant body and mind.
As for where Clark's powers went, just assume they were put in a crystal for safekeeping. That's why Jor-El/Lionel took him back to the Fortress. I mean, if Jor-El can turn back time and space itself, then he has the ability to store DNA on a crystal.
If that doesn't work for ya, just assume Clark's becoming mortal was just a plot device used by the writers. I know it's a stretch, but deal with it. lol