How to save comment name email url fields?

This is most likely side effect of your caching plugin.

WP stores this information in user’s cookies and tries to read them to fill in the form. However since it happens on server side it needs to output info into a page’s source. So if page is being served from cache it just doesn’t work.

As far as I remember full page cache plugins tend to have option to not serve cached pages to known commenters. However that obviously reduces efficiency of caching and might not be a practical option, depending on how many visitors are such.