I think one thing that will be helpful to this discussion is recognizing that our understanding of sexuality and use of terms like hetero/homo/bi-sexual are difficult if not impossible to apply to ancient peoples. Simply put, a male Greek or Roman who had sex with another man would not consider himself homosexual simply because that wasn't a concept. The emergence of such ideas is pretty fascinating, but beyond the scope of your question and of my knowledge.

With that addressed, let's take a look at your questions.

Sex was certainly part of the life of the ancient Greeks and Romans, but it was not something like wrestling or foot racing which were open activities done in the public sphere, whereas sex was a private matter. Of course, one's sex life was open to criticism and gossip or admiration.

That said, homosexual relations were very real and well-attested. The Greeks had an informal custom of pedastry wherein an older male would protect, educate, and serve as a role model for a young, beautiful, male youth. But other relationships certainly existed. This is a good debrief from course material on sexuality in Ancient Greece, including types of relationships. As another example, in Plato's Symposium, Alcibiades, a Greek general tries to seduce Socrates, offering to be the submissive partner. With the Romans, homosexual relationships certainly existed, though an element of class was key. A Roman aristocrat was expected to marry and procreate with a woman, but for much of the Late Republic period and onward, there was nothing wrong with taking male lovers of lower classes or who were slaves. Indeed, most sexual relations between Romans and between Greeks are probably better understood as power relationships than gender ones. That is, it was important to be the dominant and not the submissive partner.

Famously, Julius Caesar dealt with an accusation that he was a submissive partner to the King of Bythnia throughout his life. There is no evidence for it, but clearly it was well enough known to 1. appear in graffiti at the time and 2. make its way into a chant during his Triumph. through Rome.

To address the last part of your question, there are some recorded instances of male-male relationships, but it is not always clear if these are "life partner"-type relationships or if they were real at all. One of the best and better known examples of this would be the Sacred Band of Thebes - an elite group of 150 male couples (300 men in total), who supposedly fought harder in part to impress their lovers. This is recorded by Plutarch, but has come under criticism for its accuracy in modern scholarship. Another disputed pair would be Achilles and Patroclus from the Illiad. Homer does not depict an explicitly sexual relationship between the two, but since ancient times, different readers have read in such a relationship. It is important to note though, that Achilles had both a wife and a son.

One last note, which is not directly relevant, but is fascinating so I can't resist mentioning it, is the gender and sexuality of the Roman Emperor Elagabalus, who, according to sources, married both men and women and preferred to present himself using female titles. Cassius Dio reports that he married his chariot driver Heirocles and liked to be called Heirocles' Queen. He has been viewed by some as an early transgender figure, but that is obviously open to debate.