What would have happened if Gulf War I had not happened and we did not respond after Saddam invaded Kuwait? I think Saddam's plan was to go after or at least weaken the Saudis then re-start his war with Iran. Gulf War I made that impossible for Saddam.
Dubya's war was ill-conceived and ill-carried out, mostly due to poor planning at the top.
However, at the time, world political support for sanctions were drying up, France, Germany and Russia were queueing up to scoop up post-sanction oil contracts and Saddam was going to try to rebuild his military from nothing.
What would have happened if we hadn't invaded? I think Iraq would have rebounded and resumed its regional ambitions. I'm not sure who Saddam'd go after, given he would be well aware of what we could do to his military, but he might have tried more cover methods to get his way. Would there be a war between Saddam and one of his neighbors by now? Good question.