New Delhi, June 22 -- As Iranian media boast of the "magnificent defeat" of the United States, Tehran is pressing its advantage, racing to sell oil under the interim peace agreement signed this week and trying to halt Israeli attacks in Lebanon.

But the Islamic Republic suffered its own major blows and faces challenges ahead. Its economy is in shambles after the war, the country was rocked by mass protests in January, and its supreme leader has yet to appear in public. It is entering nuclear talks with the U.S. after coming under attack during two previous rounds.

The deal offers desperately needed sanctions relief, much of which will come only if Iran rolls back its nuclear program, including at a minimum diluting its stockpile of high...