The House has the power to fulfill this duty. The constitution provides that "[e]ach House may determine the Rules of its Proceedings." The House already uses this power to determine the roles and rules for other non-voting delegates. Seating a delegate does not require a constitutional amendment, or new piece of legislation. This treaty was already ratified. Speaker Pelosi could call a vote tomorrow to seat Teehee as the Cherokee Nation delegate and be perfectly in line with the rules of the House.
Yes. The Treaty of New Echota is a living, valid treaty. The delegate provision remains in force today because it has never been abrogated. To abrogate a treaty, Congress must do so in clear, unmistakable terms. This has not been done. The Cherokee Nation has adhered to its obligations under the treaty and is asking the United States to do the same.