Ok, the problem's fixed.
According to longhorn180's mod, you have to remove the following from the product.tpl file (lines 33 and 34 in my file):
</td>
<td valign="top" width="100%">
I was able to fix my buttons and lines by adding the </td> again a little later, after the call for the product.fulldescr, but before the table for the Product Details is created. (In my file it's at line 57) It goes
</if>
</tr>
</table>
<p /> </td>
When I added that </td>, all my troubles disappeared!
Thanks to everybody who makes this forum such a great resource!