Her job dismissal

Except that Alcoholism is considered a disease, so under most union contracts (assuming she teaches public school) she wouldn't be fired, but put on medical leave. Given that she was in treatment at the time, she could have filed for wrongful termination. I'm really surprised Nick Offerman's character didn't advise her that her situation was temporary, and to contact her union officer. Though, I suppose that might give people the wrong message that there are no repercussions...

In the US most of us live in "right to work" states. That permits employers to fire you for a reason or for no reason at all. You can be sick, you can be healthy, you can be a bad worker, you can be a good worker. Doesn't matter. Unions (especially teachers' unions) interfere with this right and it's extremely difficult to fire a teacher (and have it stick). Excepting teachers, you'd better believe that alcoholics, and others with drug addictions are rightly fired all the time. They have no legal recourse and aren't able to collect unemployment benefits either (as they caused their firing).
Let me add the NOTE: that discrimation as laid out in the law (sex, religion, race, creed, age) are exceptions to the rule that you can be fired for any or no reason, but you have to be able to prove that was the reason for your firing.

how would it be illegal to fire her? she showed up to work drunk, threw up in front of children, and then lied to her boss for months.

The reason her boss was not more lenient was way more about the lying than the fact that she was an alcoholic.

Many union teachers have been fired for drugs and alcohol. They have to provide treatment but do not have to hold a certain job.

All that aside it was in the movie to show that when you get sober it is only the beginning. If you are an arrogant selfish drunk, after you get sober you are most likely an arrogant selfish sober person. It is a lot of work.

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I don't know how I feel about this...I thought her coming clean and telling the principal the truth was the right thing to do. She only made up that story about being pregnant because one of the children mentioned it after she got sick in front of them. At that time she hadn't gotten sober so it was for her a way out. I still think the principal had no compassion for her "trouble" but apologized for how the children treated her with all those probing questions. I think those children were so out of line for putting her on the spot and didn't seem real to me. I would have told them all to keep quiet and get back to the addition. No explanation was needed from the teacher about her personal business.

Personally I think the principal told her she wasn't wanted in that school because she lied about being pregnant. Finding out she was also an alcoholic put her over the edge. If she had been perceptive the principal would have known she had some issues.

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