All of the above are fairly long literary works, and I would imagine they would take considerable time to recite them, so how was this done? And during what social circumstances? Were the epics broken up over a course of several days, or were epic never recited in its entirety but only told in smaller episodes? (E.g. "Let me tell you guys about Ulysses and the cyclope's island", or "have I told you about when Utnapishtim told Gilgamesh about how he survived the flood?")
Your guesses are spot on. These epics are broken into pretty discrete episodes, and you'll see scholars referring to "The Song of Diomedes" as a unit of the Iliad (that being the section where Diomededs leads the Greeks and wounds Venus). The rhapsodes who performed these pieces would choose an episode, or perform an episode on request.
During the Odyssey, we have scenes where Odysseus is playing a rhapsode. He tells his own story to the Phaeacians (the Odyssey begins 10 years into O's journey when he washes up in Phaeacian territory where he tells his story beginning with the fall of Troy). His story takes place over a few nights, telling one book each night. So for Homer, it was realistic to portray a rhapsode performing up to 1000 lines of a poem in one night.
For more info check The World of Odysseus by M.I. Finley. He bases his information on rhapsodes from anthropological studies of Slavic bards, but I cannot for the life of me remember the author of those. Its a pretty influential study.