Are there other whitespace codes like &nbsp for half-spaces, em-spaces, en-spaces etc useful in HTML?

Yes, many.

Including, but not limited to:

• non breaking space :   or  
• narrow no-break space :   (no character reference available)
• en space :   or  
• em space :   or  
• 3-per-em space :   or  
• 4-per-em space :   or  
• 6-per-em space :   (no character reference available)
• figure space :   or   
• punctuation space :   or   
• thin space :   or  
• hair space :   or  

span{background-color: red;}

<table>
<tr><td>non breaking space:</td><td> <span>&#160;</span> or <span>&nbsp;</span></td></tr>
<tr><td>narrow no-break space:</td><td> <span>&#8239;</span></td></tr>
<tr><td>en space:</td><td> <span>&#8194;</span> or <span>&ensp;</span></td></tr>
<tr><td>em space:</td><td> <span>&#8195;</span> or <span>&emsp;</span></td></tr>
<tr><td>3-per-em space:</td><td> <span>&#8196;</span> or <span>&emsp13;</span></td></tr>
<tr><td>4-per-em space:</td><td> <span>&#8197;</span> or <span>&emsp14;</span></td></tr>
<tr><td>6-per-em space:</td><td> <span>&#8198;</span></td></tr>
<tr><td>figure space:</td><td> <span>&#8199;</span> or <span>&numsp;</span></td></tr>
<tr><td>punctuation space:</td><td> <span>&#8200;</span> or <span>&puncsp;</td></tr>
<tr><td>thin space:</td><td> <span>&#8201;</span> or <span>&thinsp;</span></td></tr>
<tr><td>hair space:</td><td> <span>&#8202;</span> or <span>&hairsp;</span></td></tr>
</table>

There are codes for other space characters, and the codes as such work well, but the characters themselves are legacy character. They have been included into character sets only due to their presence in existing character data, rather than for use in new documents. For some combinations of font and browser version, they may cause a generic glyph of unrepresentable character to be shown. For details, check my page about Unicode spaces.

So using CSS is safer and lets you specify any desired amount of spacing, not just the specific widths of fixed-width spaces. If you just want to have added spacing around your h2 elements, as it seems to me, then setting padding on those elements (changing the value of the padding: 0 settings that you already have) should work fine.

Not sure if this is what you're referring to, but this is the list of HTML entities you can use:

List of XML and HTML character entity references

Using the content within the 'Name' column you can just wrap these in an & and ;

E.g.  ,  , etc.