Better format capability for Filtered HTML

This is not really a Wysiwyg issue since this would happen without the module as well.
The Filtered HTML text format comes with a predefined set of serverside filters enabled which are configured to only allow the most basic formatting. Drupal saves input as it was entered by the user and applies these filters during rendering of the content. Serverside filters must be applied like this since even if you are using a WYSIWYG editor, users can easily disable it and enter anything wherever you ask for input.

If you go to the Content Authoring section you can change which filters are enabled on each format and alter their settings.
Wysiwyg module isn't able to set change these settings automatically, or automatically adapt the editor settings to match them, because of the enormous amount of settings permutations and differences between editors.
You need to configure the filters to accept whichever features you have enabled in the editor. If you want users to use the <strong> element, add it to the list of allowed tags in the "Limit allowed tags" filter.
I would recommend disabling the filter which converts newlines to <br /> and <p> tags as WYSIWYG editors already know to use these when the user hits the Enter key. If you leave it enabled and the editor is set to apply source code formatting, any newlines it uses for that formatting will be changed by Drupal as a <br /> or <p> tag during rendering.

Note that many WYSIWYG editors apply formatting using style attributes on several tags (like paragraphs), and the "Limit allowed tags"-filter shipped with Drupal and enabled on Filtered HTML will always remove style attributes on all tags for security reasons. (They may contain harmful URIs.) There is a replacement for that filter called WYSIWYG Filter which allows for much more fine grained control over what Drupal considers to be allowed in the output.

Once you have changed these settings, any existing nodes using that format and having content including previously disallowed markup will automatically work.

I hope that answers your question.

Umm, what happened there?

Ok, but since you created a new issue for that, let's close this one.