AL-KHOKHA, Yemen—The United Arab Emirates paints its battle for the Yemen port city of Hodeidah as pivotal to forcing Houthi rebels to negotiate an end to the three-year war. It is also the biggest test yet of the U.A.E.’s growing military reputation.

But the U.A.E. has now suspended the advance after confronting land mines, drones, snipers and humanitarian challenges, prolonging a drawn-out campaign in a broad contest for regional domination that pits the Emirates, Saudi Arabia and other Sunni Muslim Arab nations against...