The House of Black and White says:

Bestow (10).

Dominance Action: Kneel The House of Black and White and discard X gold from it to choose and kill a character with printed STR X or lower (X cannot be 0). You may move any number of gold from your gold pool to The House of Black and White.

On thronesdb, the ruling is saying that in order to move gold to THoBaW, you must target and kill a character in order to move gold onto the card. However, there is no "then," listed in the wording. Is that correct? Does the first part need to be met in order to move gold onto it?

From the Unofficial pack FAQ, from the people that know what they are talking about:

The House of Black and White

• Killing a character and moving gold onto The House of Black and White is all part of the same action. You must legally target a character for a kill in order to move gold onto The House of Black and White.
• Moving gold does not depend on whether the kill was successful or not.
• You may kill a character with printed strength 0 (or X) by discarding 1 or more gold.

I'll admit, I'm not rules savvy enough to understand exactly why it works that way; I think it is due to the legality of using it in the first place (i.e. you must have a legal target to initiate the effect, and once you've done that the entire effect will resolve).

It's indeed because you need a target to trigger the effect. A "then" would mean that you must successfully kill the target in order to move gold (which would make it a really weak card).