The reference to a house does not make sense. It's better to just mention what the idea is rather than follow an antiquated and anachronistic convention. It really doesn't matter, as long as both sides make good arguments and follow the rules. Nevertheless, the expression is confusing and somewhat prolix.
No, debate topics should not use "This House Would" in their titles, because it is confusing and it makes it harder to debate. It is harder to debate if people involved are not clear on the topic. It's hard to know what "This House Would" means, and it only confuses a debate. Debates should be written simply so people can participate knowing what they're talking about.