A listing of all these individuals and their connections in both learning and davening can be found in "The Arizal: The Life and Times of Rabbi Yitzcak Luria" published by Hamesorah under the Artscroll Youth series.
They list in the footnotes as sources, Ta'amei HaMinhagim, pg 260, note 28, Shivchei HaAri by Rabbi Shlomo Shlumiel ben Chaim pp 37-38, Reishit Chochmah, Sha'ar HaKedusha, ch. 7 by Rabbi Eliyahu De Vidas, Shnei Luchot HaBrit, masechet Shavuot; U'lehearot by Rabbi Yishayahu HaLevi Horowitz and Shem HaGedolim, Shin, chap. 26 by Rabbi Chaim Yosef David Azulai.
Another early source is usually published with Ta'alumot Chochmah by Rabbi Yosef Shlomo Rofeh Del Mediggo.
Another excellent book recounting many of these types of stories, although not new, is Legends of Safed by Dov Silverman published by Gefen Ltd. Publishing.
The sources are purported to affirm that these individuals prayed together and learned together.
In the case of Rabbi Azikri, it lists him as the gabbai/shammash of the Beit Midrash, not the Beit HaKenneset. According to how it is explained, Rabbi Yosef Karo (Beit Yosef) headed the studies of halacha. Rabbi Alkabetz headed Nistar and was succeeded by Ramak who was succeeded by the Ari z"l after Ramak passed away.
The Ramak, the Alshich, the Ari z"l, Rabbi Azikri, the Beit Yosef and Rabbi Alkabetz had overlapping dates. The Ari z"l came late in his life to Tzfat. So the time they would all have been in one place was probably fairly short.