Heading tags are "block-level" elements in HTML, meaning that anything added after them automatically is placed on a new line.
I'm sure this is probably something that has to be resolved by changing the margin/padding for the heading tags in the CSS.
Yes, if you are trying to adjust the default spacing below headings, the best way to do that is not to add in a line break, but instead to edit your CSS for that heading.
margin: 0 0 10px 0;
Though, I am curious to know the technical reasons as to why text placed after a line break cannot be formatted individually from a heading tag placed above the line break.
Just a quirk of the WYSIWYG editor that causes the closing heading tag not to be inserted properly. I wouldn't worry too much about it - just don't add manual line breaks after headings. ;-)