What are the main sources for information about his life? Are we sure he existed?
We don't, and we aren't. The main sources for the biography are ten texts known as the "Lives" or Vitae, and which can be conveniently found in M. L. West, Homeric Hymns. Homeric Apocrypha. Lives of Homer (Loeb Classical Library 496, Cambridge, MA, 2003).
But the historical content in them is very problematic. In terms of actual things that really occurred in the past and could in principle be observed if we had a time machine, their value is basically nil. On the other hand, in terms of the contemporary understanding of the biography of Homer -- and other poets, like Hesiod, Sappho, and Solon -- they are definitive. The Lives represent a more or less authentic understanding of who and what Homer was for Greeks of the 5th century BCE.
I say "5th century BCE", rather than "7th century" (the actual date of composition of the Homeric epics), because that's when the material in the Lives seems to go back to -- quite probably the late 6th century, in fact, but there's no way of being 100% certain of that. And I say "more or less authentic" because our actual sources are much later than the 5th century BCE, but almost certainly are a fairly faithful representation of material associated with an early 4th century figure named Alkidamas, who was a key figure in a traditional story that was passed down to later centuries as the Contest of Homer and Hesiod. We have a full text of the Contest (still widely known by the Latin title Certamen, though it's in Greek) dating to the 2nd century CE or later, but significant chunks of it go back to Alkidamas; and there are some pretty good indications that it went back to the 5th century (e.g. the poetic duel in Aristophanes' Frogs apparently plays on the story). The Contest is a central text because it is directly linked to the Lives -- parts of it are normally considered to be one of the Lives -- and because it's also a key story for 5th century conceptions of the biography of Hesiod. In the 5th and 4th centuries, Homer and Hesiod were routinely treated as a pair of binary opposites: the Contest is the purest crystallisation of that idea.
But we have reliable biographical sources for hardly any individual in the Greek world prior to 550 BCE or thereabouts, and basically nothing prior to 600. Everything prior to that date is automatically suspect. If it's not a significant political/military figure, it's even more suspect. And since the tradition that does survive focuses on heavily mythologised episodes, things like a battle rap between the two most famous poets of early Greece (Homer and Hesiod), set at the funeral of a hero (Amphidamas) of a pan-Hellenic war that quite possibly isn't historical either (the Lelantine War); or a riddle that Homer couldn't solve, with the result that he got depressed and died; or other things like that, then -- well, there's no reason to attach any credence to it.
But that doesn't mean it's not important. It's representative of what the extant 5th century sources believed about the 7th century. When the Homeric epics first became truly popular in the late 6th and 5th centuries, people were already treating this legendary biographical tradition as equivalent to actual events of the past. The fact that they were wrong to do so isn't really the point: how they conceived Homer, and other poets like Hesiod, Sappho, etc., is in a very real sense more important than any actual events surrounding real individuals of those names.
For our purposes what it comes down to is that "Homer", "Hesiod", etc. are not so much names of authors in the sense of physical individuals, but rather labels for poetic heritages. A corollary of that is that anyone performing the works of these poets could adopt the poetic persona. Thus the poet of the Hymn to Apollo could adopt the persona of "Homer", describing himself as a blind poet whose works are the best of all, even though the Hymn was actually composed in the 6th century. The poet of the Shield could adopt the persona of "Hesiod", adapting bits of an earlier Hesiodic poem. And the poet of the Works and Days could also adopt the persona of "Hesiod", talking about "that occasion when I went over to Euboia to take part in a poetry competition at the funeral games of Amphidamas".
That's really what these names are: personas, labels. They might possibly have been real individuals once upon a time; but purely in terms of the poetry that survives, their real legacy is as personas that could be adopted by any poet working in the tradition of their poetry.
Further reading if you want to do more in-depth research: