Design View is not 100% reliable. At best, it is an approximation of what you can expect to see in real browsers. For best results, use Live View in Split view. That way you can the page as it would be rendered in Chrome/Safari in one panel and edit code in the other.
I question the need for APDivs as they are removed from normal document flow and thus rarely if ever required in primary layouts. Since you didn't provide any CSS code, it's hard to imagine what else is going on in your layout. But based on the HTML code you provided, if I type some text inside <div> and <td> tags in code view, I'm able to see them just fine in Design View.
<div id="apDiv4">some text here </div>
<td>some text here</td>
Nancy O.