Anyone can make a suit. (I've been on the receiving end of a frivolous one where the employee was caught committing fraud and sued anyway.) Frequently it's just cheapest for everyone to settle. A court date with expensive lawyers can cost more than a month's salary.

The law generally doesn't require a reason or documentation unless the employee is a protected class. If you are still positive with your boss or the CEO, you can usually ask off-line if you've signed something that says you won't sue.

In most parts of the US, no. Like the post says, "I don't like the shoes you're wearing today" is a perfectly legal reason to fire in an at-will state.

The only exception I'm aware of -- and why it's a great idea to get things in writing -- is in a couple of states it's possible to sue the employer if they said they'd do one thing and did another thing. Whether it's possible in your state, and whether it'd be worth the (both financial and emotional cost) of the legal proceedings, is something to talk to a lawyer about, though, and I am not one.

This exception is, by the way, the reason why employee handbooks and other company documents pretty much never explain specific procedures they'll use for correcting problems, other than a catch-all "may lead to termination". Putting a procedure in writing can lead to someone holding you to that procedure, and no well-advised company puts itself at that risk.