SOLDIERs are people of strong body and mind who are showered in mako and injected with jenova cells. Depending on how strong they abecome they are assigned a rank, either 1st, 2nd, or 3rd class, and if you improve yourself you can move up in rank.

Cloud was not in SOLDIER. Cloud was a ShinRa MP, he had no enhancements whatsoever while in ShinRa's employ. Cloud did recieve the SOLDIER treatment after Nibelheim though. Hojo took all the survivors of Nibelheim and injected them with jenova cells to test his reunion theory and then showered them in mako to make them comatose(mako poisoning is a collapse of the mind) so that the call of Jenova was all they could hear. Weak minded individuals cannot resist mako poisoning and the pull of jenova's reunion. Hojo called these tests the Sephiroth copy project, and other than Cloud they all became mindless Sephiroth puppets(they are then men in black capes).

So Cloud was not in SOLDIER, but he did get the same enhancements as a SOLDIER, because of Hojo's experimentation.

I think SOLDIERs, especially the First Class, are ShinRa's personal army and they're the ones who gonna be in-charge on missions that can't be handled by the ordinary ones (i.e. Nibelheim incident). It is revealed in CC that he was a ShinRa MP and not a SOLDIER. But he can become one, right? I mean, if the Nibelheim incident didn't happen.

It was revealed in FFVII that he was an MP. And no, he could not hve become a SOLDIER if Nibelheim didn't happen. His mind wasn't strong enough to handle mako and Jenova cells, which is why he falls into a coma when Hojo showers him in mako and why Sephiroth can so easily manipulate him through the Jenova cells in FFVII. Because ShinRa doesn't want vegetables in SOLDIER they wouldn't let Cloud become a SOLDIER, his mind simply isn' strong enough.