The bible I typically pull from the shelf is the Oxford Annotated Bible with Apocrypha (1965), which uses the RSV.
As for your second question, I would highly recommend The Text of the New Testament: Its Transmission, Corruption, and Restoration by B. M. Metzger and B. D. Ehrman. In brief, our best manuscript is generally considered to be the Codex Sinaiticus, which wasn't discovered until the 19th century. All of the best papyri (including p52, the earlist extant gospel fragment) were discovered after the KJV. In fact, the KJV translators relied heavily on the Textus Receptus for the Greek text, which was based on only a few, and not the best, manuscripts.
Finally there is the issue of language change and a better understanding of the original languages. Again, my knowledge of Hebrew isn't good enough to comment authoritatively here, but the most important scholarhip in the Greek language, as well as the best grammars and lexicons (in various modern languages, but especially German) were written in the 19th and 20th centuries. Also, many words which might be translated accurately enough into the early modern english of the KJV (e.g. the verb "will") have changed in current english.