I think there's one thing you got confused. The SOLDIER get infected by Mako energy and Jenova cells when they join SOLDIER. Zack was already a SOLDIER when Hojo did those Jenova experiments to Cloud and him, so since Zack already had Jenova cells inside him, he was immune to their effects. However, Cloud never made it into SOLDIER so the combination of his weak self-image and Jenova cells got him totally messed. This is said by Cloud after you get him back from the Lifestream:Originally Posted by SeeDRankLou
CLOUD: "You see, someone in SOLDIER isn't simply exposed to Mako energy. Their bodies are actually injected with Jenova cells... For better or for worse, only the strong can enter SOLDIER. It has nothing to do with the Jenova Reunion. But weak people... like me, get lost in the whole thing. The combination of Jenova cells, Sephiroth's strong will, and my own weaknesses are what created me. Everyone knew that. I'm... Cloud... the master of my own illusionary world. But I can't remain trapped in an illusion any more... I'm going to live my life without pretending."
I don't think Cloud got rid of Jenova cells after getting out of the Lifestream. He just found himself and restored his self-image. Besides, the Jenova you had chased through the entire game (the one who looked like Sephiroth, and whose Birth, Life and Death forms you fought) was destroyed and the real Sephiroth was in the Northern Crater, sleeping and waiting for the Meteor to hit the Planet. The Reunion wasn't going on anymore, so neither Sephiroth nor Jenova were trying to control those with Jenova cells anymore, so that's why Cloud doesn't go berserk for the rest of the game.
If you remember, Cloud hears Sephiroth's voice in his head once more, after the final battle. He gets messed up one more time, and inside his mind, he Omnislashes Sephiroth to shreds. However, after this, you see both red and green lights going inside Cloud. Red indicates Jenova, green indicates Holy/Lifestream. The screen blinks because of those lights, and after a while only green energy flows out of Cloud, and there's no signs of Jenova's red energy anywhere. You could interpret this in many ways. However, I always think it so that this is where Lifestream cleans Jenova cells out of Cloud for good. It also works with the symbolic meaning that Cloud destroyed Sephiroth (who symbolizes Jenova) in his mind to be free of those evil cells.
BTW I still have doubts about how Zack can die if a few Shinra guards shoot a few bullets through his chest. Cloud and Tifa got Sephiroth's huge sword impaled through their hearts (and Cloud didn't have Mako or Jenova cells that time, he was just a normal guy) and they still survived. Cloud killed Sephiroth, but within five years, Jenova revived him. Now why wouldn't it be possible that the Jenova cells inside Zack restored him too, though he was still messed up because of the damage, so he ended up sick in the pipe in Midgar?