Blockquotes don't show a closing quote

Looking at your screenshots, this appears to be a CSS issue. Your theme is probably showing one background image as the opening quote and doesn't have anything for the closing quote.

This is not an issue with Wysiwyg or CKEditor but your theme.

I don't know what your CSS looks like so I can only guess, but I think I understand what the problem is.

The blockquote effect used here on Drupal.org only uses a single background image for the large opening double qoute character, like rocketeerbkw said. I suspect your theme is using the extra <span>&nbsp;</span> tag at the end of your blockquote to put a second background image in, for the large ending double quote character.

CKEditor does not know it needs to put that extra span tag in there, so your theme does not have anything to put the second background image on. To make CKEditor do that, you would need a plugin that would make sure the extra span is always there.
I don't know of a plugin like that, but if there is one, you could create a small Drupal module to tell Wysiwyg module that plugin is available for CKEditor, using hook_wysiwyg_plugin().

This will however always put the extra spans on all blocquotes in your content, something which may be undesirable if you one day you decide you don't need that extra span. Another solution would be to create a Drupal module implementing a text filter to do this instead. A text filter could automatically add the span tag inside all blockquotes it sees (for content using a text format with that filter enabled) during rendering, so the span would not actually need to be stored with the content in the database. The downside would be that you would not see the ending quote inside the editor because it can not use the text filter system.

I lost track of this issue, apologies for that.
I'm going to close this issue to clean up the queue.
If you still have questions about implementing hook_wysiwyg_plugin(), please re-open it.