[Son of Númenor]

But I'm curious as to how many times he can be killed?

As I said, there is no set amount.

It's like he has constantly renewable forms he can take, but there comes a point when he can no longer assume a fair form, such as after Numenor.

The only explanation is that the number of times he could assume another form before being unable to take a 'fair form' is subject to the whim of Eru.

If that's so, how did the Ring's demise cause him to lose form?

The Ring was tied to him. When the Ring was destroyed, he lost the power to have a physical form or a 'potent spirit' (to misquote the movies) within Middle-earth.

Where did his spirit go?

What remained of Sauron's 'spirit' went into the Void with Morgoth. The part that was destroyed with the Ring was, um, destroyed.

Do mortals have the power to thrust Maia into the Void?

No.

I hope that helps.