Are there leviyim in the Beta Israel Community? If so, how does their role in the Beta Israel Community differ from other Jewish Communities? (Do they get the second aliyah, etc.)
I want to preface my answer by noting that I am very far from an expert or authority on the Beta Israel or their traditional minhag (known as haymanot).
However, from what I have been able to understand, there are no levi'im (in the genealogical sense) among the Beta Israel (nor kohanim, for that matter). The main clergy of the Beta Israel, the qessoch (sing. qes), are also known by the title kahen, and do claim priesthood from the Aaronic lineage, although it is not a hereditary title. They are served by debteroch (sing. debtera), who were a combination of teachers, healers, cantors and musicians (since Beta Israel services were accompanied by instruments), so in that sense they might be analogous to levi'im.
The traditional Beta Israel liturgy was very different than the liturgical cycle of other Jewish communities, and there was no weekly ritual reading of the Torah (and therefore no 'aliyot). With the arrival of European, and eventually Israeli, Jewish educators, Beta Israel religious life underwent massive changes to align itself more closely with other branches of Judaism, and of course this process has only intensified in Israel. I know there are some Beta Israel synagogues emerging in Israel today and would be very interested to hear how they run their prayer services, and what they do with the Torah reading, aliyot, etc.
To learn more about Beta Israel liturgy, prayer, and religious life, check out these scholars (among others): Kay Kaufman Shelemay, Margaret Hayon, Frank Alvarez-Pereyre and S. Ben-Dor, Olivier Tourny, and Ron Atar.