I can't answer your first question, unfortunately.

To comment on your second question, the former ruling dynasty in Ethiopia (which ended in 1974 with the deposition of Haile Selassie) claimed direct lineage from Menelik, who was purportedly the son of king Solomon of Israel and the Queen of Sheba. This tradition dates as far back as the thirteenth century A.D. and is recounted in the Ethiopian national epic The Glory of the Kings. However, the notion that the Queen of Sheba came from Ethiopia apparently stems from a conflation of the "queen of the south (i.e., Sheba)" in Matthew 12:42 and Luke 11:31 with "Kandake queen of the Ethiopians" in Acts 8:27.^1 Nothing in the biblical account of Solomon and the Queen of Sheba (1 Kings 10:1-13) suggests that the two engaged in sexual relations.^2

The historical Sheba (Saba) actually lay in South Arabia (Yemen), and some ancient Arabian women are known to have exercised power or at least a degree of independence.^3 Nevertheless, the historicity of the tale of the Queen of Sheba remains impossible to determine. What can be said is that trade between Arabia and the Levant may have already existed in the time of Solomon (tenth century)^4 and perhaps even earlier.^5

Hopefully someone else can offer you a fuller response! :)

 

Footnotes:

^1 Witold Witakowski and Ewa Balicka-Witakowska, "Solomon in Ethiopian Tradition," in The Figure of Solomon in Jewish, Christian and Islamic Tradition: King, Sage and Architect, ed. Joseph Verheyden (Leiden and Boston: Brill, 2013), 219-40.

^2 Marvin A. Sweeney, I & II Kings: A Commentary, OTL (Louisvile: Westminster John Knox Press, 2007), 149-51.

^3 See e.g. Israel Eph'al, The Ancient Arabs: Nomads on the Borders of the Fertile Crescent, 9th-5th Centuries B.C. (Jerusalem: Magnes Press, 1982), passim; Kenneth A. Kitchen, "Sheba and Arabia," in The Age of Solomon: Scholarship at the Turn of the Millennium, SHCANE 11, ed. Lowell K. Handy (Leiden, New York, Köln: Brill, 1997), 126-53; André Lemaire, "La Reine de Saba à Jérusalem: la tradition ancienne reconsidérée," in Kein Land für sich allein. Studien zum Kulturkontakt in Kanaan, Israel/Palästina und Ebernâri für Manfred Weippert, OBO 186, ed. Ulrich Hübner and Ernst Axel Knauf (Freiburg: Universitätsverlag; Göttingen: Vanderhoeck & Ruprecht, 2002), 43-55, esp. 50-2.

^4 Mario Liverani, "Early Caravan Trade between South-Arabia and Mesopotamia," Yemen 1 (1992): 111-4; John S. Holladay, Jr., "Hezekiah's Tribute, Long-Distance Trade, and the Wealth of Nations ca. 1000-600 BC: A New Perspective," in Confronting the Past: Archaeological and Historical Essays on Ancient Israel in Honor of William G. Dever, ed. Seymour Gitin et al. (Winona Lake: Eisenbrauns, 2006), 310-31, esp. 314-21.

^5 Michaël Jasmin, "Les conditions d'émergence de la route de l'encens à la fin du IIe millénaire avant notre ère," Syria 82 (2005): 50-60; idem., "The Emergence and First Development of Arabian Trade Across the Wadi Arabah," in Crossing the Rift: Resources, Settlements Patterns and Interaction in the Wadi Arabah, LSS 3, ed. Piotr Bienkowski and Katharina Galor (Oxford: Council for British Research in the Levant and Oxbow Books, 2005), 143; Caroline Grigson, "Camels, Copper and Donkeys in the Early Iron Age of the Southern Levant: Timna Revisited," Levant 44 (2012): 82-100.