Translation:He is reading a book.
书 does mean book specifically. There are more precise words for journals/diaries etc. 看书 can also mean studying though. 看 is a verb also used to watch things such as films and to look at something. I think of it more as "to consume with one's eyes" instead of read/watch. Unless you already mentioned a book I'm not sure if in a real conversation it would be clear what he was doing without using 书
No: in English, "he is reading" in no way implies he is reading a book; rather, a book is just one of many things he could be reading.
In fact, in English, "reading" also applies to activities that have nothing to do with words on a page: a doctor "reads" your temperature, a fortuneteller "reads" your future, a psychic "reads" your mind, poker players "read" their opponents' betting patterns and reactions.
In fact they both mean "read(ing) a book" or "read(ing) books", though "读书" is also often used to mean "study".
(I believe "看书" can also be used to mean "study", but in a more limited sense, as in to be currently sitting at one's desk and reading one's books, but I'll have to defer to native Chinese speakers on this point.)