I just want to confirm that I am reading the mission right.

It says the Threat Level is equal to the round. So say according to the other rules the threat starts at 4. Then the round after the Imperial Player gets, 5 threat then 6 etc.

Is this kind of a soft turn counter as there is not a set round limit... but if the Rebels take too long they will get overwhelmed by the threat?

Also there are some characters in the closed groups that only get deployed on certain conditions. Now if they do not get deployed based on that condition can you then deployment. I ask because I want to use some of the closed groups.... so it would suck if they get completely locked from me.

Yes, the increase threat step gives you the amount of threat that the round counter shows.

If reserved groups do not get deployed by the events, they do not get deployed. They are still reserved groups, not your open groups. You don't get to use them.