One or possibly several arks probably existed in Iron Age Palestine. These would have been portable shrines that could be used for divination or carried out into battle as a talisman. (Both uses are portrayed in the Old Testament.) Similarly, Assyrians would bring idols on chariots into battle, and pre-Islamic Arabs carried a shrine called a qubbah into battle.

Most scholars think the ark in Jerusalem was taken or destroyed by the Babylonians when the city was ransacked in 597 BCE. It's not impossible that Pharaoh Shoshenq took it earlier during his 10th/9th-century conquests in Canaan, as is presupposed by the plot of Raiders of the Lost ark. The ark disappears from the biblical narrative once it is ensconced in Solomon's temple.

No ark existed in Herod's temple, and whatever the Ethiopians have, it is certainly not anything that belonged to the ancient Israelites or Judahites. I'm vague on the details, but the Ethiopian church venerates certain altar tablets called tabotat (singular: tabot) made of various materials (wood, metal, etc.), analogous to the Coptic maqta' used as a tray for holding the eucharist wine and bread. In Medieval times, the Ethiopian tabot was shaped more like a box and could be used as a portable altar for church, and thus fit the description of an ark — which just means "box". In the 15th century, King Yaqob of Ethiopia compared the Ethiopian tabot symbolically with the womb of Mary and the Mosaic Law, and exaggeration of this symbolism over the following centuries resulted in the mistaken belief that the Ethiopian tabot actually was the Ark containing Moses' law tablets. The Ethiopians have done nothing to dissuade people of this misconception, and the most sacred tabot in Aksum has therefore come to be equated with the Ark of the Bible.

besides church elders

The concept of "elders" in ecclesiastical polity is a Protestant one (a somewhat uncharitable assessment could be that people wanted to come up with a good equivalent of the Greek "presbyteros" but refused to use the most direct translation, the etymologically-descended word "priest," on the grounds that it sounded too Catholic). The Oriental Orthodox churches, such as the Ethiopian Orthodox Tewahedo Church, maintain the same kind of polity as the Eastern Orthodox, with deacons, priests, and bishops, with the main difference from the Catholic polity being that there is no bishop who is above the others (though some are more senior than others and there is a patriarch who ranks first among equals). Therefore, there are no "elders" in the Ethiopian Orthodox Tewahedo Church.

Only one priest is allowed into the building where the Ark is kept. Therefore, it cannot be proven to a scholarly standard of evidence and must be accepted on faith by members of the church.

Further question: I would like to know if the claim to the Ark has any connection to the fact that the Ethiopian Orthodox Tewahedo Church seems to maintain more ancient Jewish customs than most churches.

Edit: Punctuation