I don't believe there's really an exact equivalent to the editions you mention. With biblical texts the "editions" are the different translations, which you'd normally find as complete Bibles, rather than as separate books with editorial apparatus as you would for other classical texts. So rather than looking for an edition with introduction and notes etc., you'd consult one of the standard biblical translations for the text and then look at another book for the equivalent of introduction and notes.
The usual translations favoured by scholars are the RSV/NRSV and perhaps also the REB and Jerusalem Bible. For the secondary literature, obviously there's a vast amount. Cambridge University Press has a very good series called "New Testament Theology", with each book focusing on just one text (or group of texts) of the New Testament, and these do pretty much the job you describe. It has another series called (amazingly) "Old Testament Theology", none of which I've read, but I would guess that they'd be much the same for those texts (and they seem to be more recent too).
There are undoubtedly other, similar series to these as well, but I'm not familiar with them. If you look in the theology section of your university library I'm sure you'll find shelves devoted to the various biblical books, and you will probably see similar-looking books appearing in each section. Those would be the ones to grab.
Thank you!