How can one reconcile the Hadith: "The Hour will not be established until the House is not pilgrimaged to" with the Hadith: "The House will surely be pilgrimaged to and 'Umrah performed after the emergence of Gog and Magog"?

The two mentioned hadiths are authentic. A reconciliation between them is possible in that Hajj might become impossible at a certain time near the Hour's advent. Nevertheless, this does not necessitate that people will not perform Hajj after the emergence of Gog and Magog. "People will surely perform Hajj to the House" refers to the location of the House, because it will be destroyed and will not be rebuilt thereafter.