Translation:The birds are cleaning the hippo with their beaks.
In French, the singular their, "leur" is used if each of the owners has one of them. Each bird has just one beak, so even though it is multiple birds, it is "leur bec." You would use "leurs" in a case where each owner had multiple items. For example: Les hommes aiment leurs casquettes. / The men like their caps. Each man in this example has multiple caps that they like.
Nope. Different languages have different grammars. In English, it's "their beaks" because there are more than one beak (due to there being more than one bird), but in French, it's "leur bec" because each bird possesses one beak. "Leur" can either mean that all entities are sharing a single object, or that there is one object for each entity (the latter being true of beaks).
In English most people say "hippo" instead of "hippopotamus". Is the same true in French? Is there a more common word for "hippopotame"?
"...with their beak" should also be accepted, since it is used in English by respected writers. If you search "with their beak" in Google (in quotes, to get exactly this phrase in the results) you get 63,400 examples of it being used. They include The British Library website: "Some birds don't need to use their vocal chords to make functional sounds. Woodpeckers, for example, drum with their beak." And from US National Institute of Health: "Within the anomaly, the effect was somewhat beneficial: the pigeons with their beak anesthetized had longer vectors and were significantly..." The Times of London says: "Robins and other small birds sit on a twig with their beak open, visibly panting." The website of the publisher DK says: "Most birds, except for parrots and birds of prey, such as eagles and falcons, catch and hold their food with their beak, or bill, alone." There are lots of other examples. I have reported it. Also, please note that I am not arguing that "beaks" is wrong - it is right. I am just arguing that "beak" is also right, as shown by its use by respected sources in English. In English, you can do it either way, and either way should be accepted. Also, before people say that some phrase isn't used or isn't correct in English, they should take the time to do a Google search and see who is using it.