[Solved] First child does not work with my selector

That is because first-child(2) is not a function so it will give an error.
It should be:

$('dd:first-child, dd:eq(1)').remove();

or

$('dd:eq(0), dd:eq(1)').remove();

or

$('dd:nth-child(1), dd:nth-child(2)').remove();

This might also work:

$('dd:lt(2)').remove();

lt selector

Fiddle for lt(2)

Edit:

To delete the <br> you can do:

$('dd br:lt(2)').remove();