The Middle Ages are a broad and ill-defined period, and the Middle East is similarly broad and ill-defined in a geographic sense. So I'll take these questions one at a time, and try to treat the various cultures of medieval Islam.

Did medieval Middle-Eastern monarchs have crowns?

It's complicated. The most common Arabic and Persian word for crown, tāj (تاج), comes from Middle Persian where it was used as a symbol of Sasanian royal power. The association between the crown and royalty continued through the Islamic period—as, for example, in al-Jahiz's Kitāb al-tāj, a collection of Sasanian royal traditions. And in the Shāhnāma, a 10/11th-century epic celebrating the heroes of pre-Islamic Iran, the tāj is an emphatic mark of royal power—which is why Rustam, a major figure in the first part of the work, bears the epithet tājbakhsh, or "crown-bestower" (referring to his role as a champion of the realm).

However, early Islamic writings also tended to denigrate the crown, emphasizing the turban in its place. This attitude is best encapsulated by the hadith that "turbans are the crowns of the Arabs" (Bayhaqī, Shuʿab al-Īman), though M.J. Kister's article lists a number of other examples in which crowns are portrayed as attributes of specifically non-Arab and non-Islamic kings. If turbans were associated generally with Islam, they were especially associated with Muhammad, who wears one in pretty much every medieval Islamicate picture of him—many of these pictures, incidentally, have other figures wearing crowns, so again, you see the ways in which different modes and types of authority were represented in different ways. (Robert Dankoff's essay, linked below, has quite a few of these pictures embedded if you're interested in that.)

This distinction between turban and crown seems to have been pretty generally maintained up to the ʿAbbasid period [ED: according to Dankoff and Björkman—Richard Ettinghausen has argued for the crown's place in a three-hat system (crown, turban, tall conical hat) used by the Umayyad caliphs], where a tāj al-khalīfa was included among the caliphal "insignia of sovereignty" (ālāt al-mulūkiyya). It has been suggested that in this as in many other aspects of ʿAbbasid culture, we should see a resumption of Persian practices. Later rulers in the ʿAbbasid domains—the Buyids, the Seljuqs, and the Ghaznavids—all seem to have worn crowns for ceremonial purposes, at the very least; again, we see this more commonly in the Persianate world than in other contexts. Elsewhere, however (for example, in Spain), the crown retained its association with non-Arab and non-Islamic rulership.

Sources:

Robert Dankoff, "Turban and Crown: an Essay in Islamic Civilization" [Academia.edu link]

W. Björkman, "Tādj," in Encyclopedia of Islam, Two

M.J. Kister, “‘The Crowns of this Community’ … Some Notes on the Turban in the Muslim Tradition,” Jerusalem Studies in Arabic and Islam 24 (2000), 217-45 [link]

"Regalia - I. Crowns," in The Grove Encyclopedia of Islamic Art and Architecture [link]

Elsie Peck, "Crown iii. On monuments from the Islamic conquest to the Mongol invasion," in Encyclopedia Iranica [link]

A. Shapur Shahbazi, "Crown iv. Of Persian rulers from the Arab conquerors," in Encyclopedia Iranica [link]