They obviously fly at different speeds and to different places, so why do they almost always arrive at breakfast? Several scenes show hundreds of them arriving at the same time, so it's unlikely to be a coincidence.
We know that the Daily Prophet delivery owls always arrive at breakfast time regardless of where they're delivering to, as seen early on in Philosopher's Stone when one arrives at the shack to deliver Hagrid's copy. This is in keeping with Muggle post, which (at least in the UK) is almost always delivered in the morning.
It's a reasonable assumption, then, that any other delivery owls are redirected to the Owlery upon arrival, and are only allowed to make their deliveries at breakfast time, once the Daily Prophet owls arrive. That way, all the deliveries are made in one go, instead of having owls coming and going constantly and potentially disrupting lessons.
There's also the added convenience that, during breakfast, everyone's guaranteed to be in the Great Hall, so the owls don't have to fly all over Hogwarts in search of whoever it is they're delivering to.
I don't believe there is any canon answer for this. But I assume that the owls have been deliberately scheduled to arrive at breakfast. And I think that that is a highly reasonable time. Consider: All of the students are gathered together in one central location, so the owl visits can be well-managed, plus the hall has a high ceiling reducing the crowding effect of so many owls arriving at once. The students have just woken up and so are relatively fresh-minded, and not yet distracted by coursework, so they can pay better attention to the contents of their letters, and have a whole day to absorb them. Arriving at lunch is less appropriate because students likely have less time between classes and are more absorbed in the learning process; dinner even less so since everyone is unwinding after the daily stress.
Also, where else could the owls deliver their letters to? It is certainly a bad idea to allow them to disrupt classes in progress; it would be chaos having owls try to find their delivery targets in the halls between classes, and the dormitories are inaccessible (how would an owl give the Fat Lady of Gryffindor tower the password??).
Of course, one could ask why Hogwarts doesn't provide proper mailboxes. I think this can be chalked up to 'the magic community doesn't do things the muggle way' and 'J.K. Rowling wants readers to find the wizarding world charming'.
Breakfast delivery was the usual custom (as others have said, probably due to the nocturnal nature of many owls, the need for owls not to be seen by Muggles in daylight en route, and the customs of Muggle post). But it is not an absolute rule. There are numerous exceptions, which can usually be attributed to urgency, secrecy, or unanticipated delay. In Order of the Phoenix, for example:
- [urgency] when Harry Potter casts a Patronus in Little Whinging (the Muggle settlement where his aunt and uncle live), multiple letters arrive at 4 Privet Drive from Mafalda Hopkirk (two official Ministry of Magic communications, the first expelling Harry from Hogwarts for underage magic in a muggle area, the second rescinding the expulsion pending a disciplinary hearing), Arthur Weasley, Sirius Black, and Albus Dumbledore (howler addressed to Petunia Dursley) in the late-evening, much to the chagrin of Vernon Dursley (appropriately, the chapter is entitled 'A Peck of Owls');
- [secrecy] Hermes delivers a letter from Percy Weasley to Ron Weasley in the late-evening in the Gryffindor common room -- in the letter, Percy states that he made a point of not having it delivered in the usual morning slot for reasons of privacy (he did not want Harry Potter to see the letter, because it was encouraging Ron to distance himself from Harry... of course, Ron almost immediately showed Harry the letter anyway); and
- [unanticipated delay] Hedwig delivers a letter from Sirius Black to Harry Potter during a morning lesson (History of Magic) because the owl was attacked and probably intercepted (i.e.: Hedwig intended to deliver the letter at breakfast, but was delayed unexpectedly), probably at the behest of Umbridge.
Owls not disrupting sleep
The owls are naturally nocturnal beings and are probably forbidden from entering the living quarters during sleep time. The next opportune moment is the morning.
Even if they are expected to travel day and night with a normal distribution this will have around 9/24th of them arriving in the morning. Add this up with the subscription deliveries mentioned in F1Krazy's answer and you have a flood of owls at breakfast.