I would hope that the employer is forbidden from breaching your privacy. And, honestly, you shouldn't just cave "because it's a business relationship". I call bullshit on that. Being an employee does not make you a slave to your manager / the business owners. You remain a human being with all the rights that go with that, even as an employee. (This really should go without saying, but it seems like in this day and age, it needs to emphasized again.)

Now, they may have the right to fire you because you checked private email during work hours. But even if they have that right, you should still be able to sue them for compensation if they violate your privacy by reading the private mail you accessed.

At least, that's the "how it should be", from an ethics point of view. Obviously, how it actually is according to the law is going to depend on which country you're in. The differences are surprisingly large.

That is basically how it works, yes.

They can record anything you do if they meet certain guidelines, they can search your desk and anything you bring into the building assuming they have some basic standard written policies and are careful when the lawyers write up the details.

Laws about recording the details of any web page that was served up are highly variable. It is much like an envelope: The address on the outside of the envelope is public, the contents are generally considered private. Laws about accessing the payload are highly variable; some places require no notice, others require some notice, still others generally prohibit it.

The company can fire you for using company equipment for checking your gmail account if it violates their written policies. The web page addresses are public (they must know them to look up the IP address and get the right data served to you). They can record that, log it, and use it as justification to terminate employment.

They cannot use the password they caught in the transmission to open your gmail account; that requires a court order else it would be fraud.