In The Odyssey, how do Odysseus and his companions expect to be treated by the Cyclops?
In one word, Odysseus and his companions expected hospitality. The Greeks esteemed "xenia" very highly. This was a kind of ritualized hospitality, especially governing the relationship between guests and hosts. That's one of the reasons why Odysseus got so mad at the suitors, by the way: they abused that relationship in the extreme. Odysseus, though, expected to be given food and shelter, as he would do for a traveler far from home. Instead, he and his men got trapped, eaten, etc. This was a terrible infraction of the customs of Greek society, one that would anger Zeus himself.
This is one instance in which the fearfulness of Odysseus's men's would actually have served them better than their leader's bravado and courage. When they arrive at the island of the Cyclopes, Odysseus's crew advises caution; they "pressed [him] strongly" to take some food and a few animals (to eat later) and return to the ship. Odysseus ignores their entreaties, however, opting instead to remain and wait for the owner of the cave to return. He hopes this clearly prosperous owner will "offer gifts" to him. The reason he might expect this is the same one that other commenters have mentioned: the Greek notion of hospitality, called xenia. If it were within his means, not only should a host offer the best food and drink he has to a traveler, but he is also supposed to offer a guest-gift as well, and this is what Odysseus hopes will happen.
In The Odyssey, the concept of "xenia," the moral code involving a civic duty in which any traveler or visitor can always rely on a friendly reception from any host, has often served Odysseus well, although he has had his fair share of hardship and has lost men when he angered the gods.
In some instances, he has lingered far too long. Calypso wants to keep Odysseus with her and immortalize him. She only lets him go because she is ordered to do so. Later, when Odysseus reaches Aeaea, some of his men are turned into swine by the goddess Circe after eating and drinking their fill. Furthermore, on the island of the Lotus-Eaters, any man who eats the flowers forgets his duty to return home. Even back home, and unknown to Odysseus at this point, civil order is not being upheld and this will have disastrous consequences when Odysseus does, in fact, reach his home only to witness the treachery of the suitors, whereupon he will kill them.
Odysseus is fascinated by the land of the Cyclopes and observes them from a nearby island until he feels compelled to investigate personally. He is aware that the monster may be inhospitable and not receive him willingly; he may have no respect for "right nor law." On the island, the "huge monster" is away tending his flocks, giving Odysseus time to look around. His men encourage him to take from the vast stocks but still Odysseus wants to test the boundaries and, despite all the visible warning signs and his men's pleas, Odysseus waits to see if the Cyclops will give him "a present," as would be customary in Odysseus's world. It will soon become apparent to Odysseus that it was a mistake to expect such a hideous being to uphold the principles of decency and the "wretch" laughs at Odysseus's reminder of the rules of hospitality. Odysseus will have to use all his cunning to escape.