Hi W+D
Welcome to the forum.
You might want to have a look at the following FAQ which explains where everything is;
www.benefitsandwork.co.uk/guides-for-claimants/faq/forum.
No, you do not have to tell them your house is bought, they will ask you if you pay rent, as you don't pay rent, if you are eligible to claim the work allowance you would get the high rate (£673).
Gary