I have never seen these anchors cause any of the issues you describe, so that's why I expected this to be a CSS issue.
They are injected to allow deep-linking to modules on a page (which is regular HTML technique).
This link will take you directly to the news module in the DNN homepage for instance:
http://www.dotnetnuke.com/#8601
I did a quick test on a skin and with this:
a{background:#f00;} > Anchors not visible
a{background:#f00;display:inline-block;padding:5px;} > Anchors as little red blocks
a:link, a:visited{background:#f00;display:inline-block;padding:5px;} > Anchors not visible
So I'm guessing that in the skins CSS there is a padding or something alike for all anchors, which should be for a:link, a:visited
The file you would have to edit is most likely Skin.css and find where the background color is injected for the anchors.
BTW, even if you would remove the DNN injected anchors (you can't), this would result in any anchors injected in the text editor for deep linking to pop-up too..