Select odd even child excluding the hidden child

Pseudo-selectors don't stack, so your :not doesn't affect the :nth-child (nor would it affect :nth-of-type etc.

If you can resort to jQuery, you can use the :visible pseudo-selector there, although that's not a part of the CSS spec.

If you're generating the HTML and can change that, you can apply odd/even with logic at run-time, eg in PHP:

foreach ($divs AS $i => $div) {
    echo '<div class="box ' . ($i % 2 ? 'even' : 'odd') . '">x</div>';
}

Even trying to do something tricky like

.box[class='box']:nth-of-type(even)

doesn't work, because the psuedo-selector doesn't even stack onto the attribute selector.

I'm not sure there's any way to do this purely with CSS - I can't think of any right now.

Here's a CSS-only solution:

.box {
  background: orange;
}

.box:nth-child(even) {
  background: green;
}

.box.hidden {
  display: none;
}

.box.hidden ~ .box:nth-child(odd) {
  background: green;
}

.box.hidden ~ .box:nth-child(even) {
  background: orange;
}

<div class="wrap">
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box hidden">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
</div>

:nth-child() pseudo-class looks through the children tree of the parent to match the valid child (odd, even, etc), therefore when you combine it with :not(.hidden) it won't filter the elements properly.

Alternatively, we could fake the effect by CSS gradient as follows:

.hidden {display:none;}

.wrap {
  line-height: 1.2em;
  
  background-color: orange; 
  background-image: linear-gradient(transparent 50%, green 50%);
  background-size: 100% 2.4em;
}

<div class="wrap">
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box hidden">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
  <div class="box">xx</div>
</div>