Is it alright to have the logo without the Equal Housing Lender written underneath. Can you just have the house with the equal sign in the middle?

Not an expert, but I say no. Regulation https://www.fdic.gov/regulations/laws/rules/2000-6000.html

§ 338.3 Nondiscriminatory advertising.

(1) With respect to written and visual advertisements, this requirement may be satisfied by including in the advertisement a copy of the logotype with the Equal Housing Lender legend contained in the Equal Housing Lender poster prescribed in § 338.4(b) of the FDIC's regulations or a copy of the logotype with the Equal Housing Opportunity legend contained in the Equal Housing Opportunity poster prescribed in § 110.25(a) of the United States Department of Housing and Urban Development's regulations (24 CFR 110.25(a)).

I agree, both the wording and dog house are required.

If you read the Fair Housing Act, it stats the "logo" IS the picture and words.

The only place you can separate the two is in an oral ad in which you speak the words.

No. The logo includes both.