Was this Theodore Roosevelt's home?

No. His home in 1901 was Sagamore Hill, located in Oyster Bay, Long Island, NY.  Sagamore Hill is still preserved by the National Park Service.  The Theodore Roosevelt Inaugural National Historic Site is where Theodore Roosevelt was inaugurated as the nation's 26th President on September 14, 1901, following the assassination of President William McKinley. At the time, it was the home of Ansley Wilcox, a friend of Roosevelt's.

Were Theodore and Franklin Roosevelt related?

Yes, but distantly. They were fifth cousins to each other. However, Franklin's wife, Eleanor, was Theodore's niece (the daughter of his brother Elliot). This makes her more closely related to Theodore than Franklin was. It also makes her a fifth cousin, once removed, to her husband Franklin. Click for more on Franklin and Eleanor.

Who was Theodore Roosevelt's vice-president?

In 1901, he didn't have one. There was no provision for filling a vacancy in the vice-presidency, so he finished William McKinley's term without a vice-president. When Roosevelt was elected to a new term in 1904, his running mate was Senator Charles Fairbanks of Indiana, who served as vice-president during TR's second term.