I’m wondering how to embed Discourse community website in wordpress so that Discourse will have the same Header and Footer of WP?
The whole community in the meaning Discourse looks exacly same as WordPress including everything on header- and footer-sections? Easily you can’t. And it is quite difficult even hard way.
But you can use themes to give impression and feeling of same site/brand and including manually links to header and footer using proper theme components in Discourse
As stated, it’s not easy to create a perfect match, but there’s a theme component that can help with creating a matching header: