Hi Horus,
Good questions ...
1/ My understanding that it is not uncommon for the older eaglet to kill the smaller one - will this happen to the peregrine falcon chicks? (is it called eyass?).
It isn't uncommon, though it isn't common occurrence either. And there are some other raptor species that do it as well when food is short. As a rule, peregrines don't practice kill and eat their siblings (siblicide) in the nest. Has it never occurred? I only know of one report and it was by eminent peregrine researchers who were reporting that they had found a very weak chick in a nest with 3 other malnourished but somewhat stronger chicks that appeared to have been pecked at by its nestmates (this qualifies as aggression). The nestsite is one they are familiar with and has had an "ok" record for small clutches (1-2) but not for larger clutches (3-4). They removed the chick but it died within a couple of hours despite careful (and very experienced) handling. Their conclusion is that this was an isolated case of starving chicks being aggressive toward the weakest of their clutch and while this is technically siblicide, it is not the same as for eagles, some owls and other raptor species. Their findings show that well-fed chicks (and ours definitely are!) show little aggression towards one another and that for peregrines siblicide isn't an issue.
2/ Is the peregrine eat fish, too?
Fish is not part of the peregrine's diet, but I can't say that a peregrine or two that lives near water hasn't tried/eaten fish at one time or another. Peregrines aren't built to catch fish, their feet are just the wrong design - check out an osprey's foot and feathers which are designed for fish hunting. The peregrines we have here prey on birds, mostly birds between the size of a pigeon and a duck. They used to be called "duck hawks" because they were such common visitors to wetlands and predators of waterbirds and shorebirds.
Hope that helps,
TPC