Is there a standard version of the Bible, compared to which, all other versions would be considered either translations or paraphrases? If yes, when was the standardization done? And how common is it to study that Bible? And is it accepted across different denominations of Christianity?
The standard resides in the original languages of Hebrew, Aramaic and Greek. While there are, admittedly, small variances in the Greek manuscripts (New Testament) in particular, these variances are completely insignificant, consisting mostly of spelling variances for proper names, word order, and very slight verb tense differences, which happen to translate the same into English anyway.
Most advanced students of the Bible will learn Greek at least and oftentimes Hebrew as well and reference the original languages. My pastor translates each passage from scratch whenever he preaches on a New Testament text.
There isn't any one standard copy of the Bible. There are several copies of the books of the Bible primarily in Hebrew, Greek, and Aramaic that are used together when translating the Bible. Which books are canonical was decided over several centuries by different groups, a process that wasn't finalized until well into the 16th century. Each part of the Bible was standardized at a different time.
The oldest established section are the 5 books of the Torah. They were canonized sometime before 444 BCE but possibly around the 7th to 6th century BCE. The prophets - Joshua, Judges, 1 & 2 Samuel, 1 & 2 Kings, Isaiah, Jeremiah, Ezekiel and 12 more - were also established sometime around the 4th century BCE. There's no agreed upon date for when the remainder of the Old Testament books, called the writings, were established for Judaism. They were still disputed as late as the 2nd century CE. Some of them were included - Ruth, Psalms, Proverbs, Job, etc - while others were not - Tobit, Judith, Macabees, etc. However they were still in the Greek translation of the Hebrew Bible, the Septuagint, that was used widely by early Christians.
The New Testament was compiled over a few centuries and was informally agreed upon as early as the 4th century CE but there wasn't a formally approved canon for Roman Catholics until the Council of Trent in 1546. Trent included 45 books from the Septuigant, the 37 in the Hebrew Bible and 8 deuterocannonical books as the Old Testament plus 27 books in the New Testament. Martin Luther's translation listed the deuterocanonicals under a separate heading in 1543 and the Westminister confession excluded them completely in 1647.
It is important to keep in mind that the Bible is a collection of books and not one single book. Therefore, there are standard versions of the bible per denomination..
There have been many attempts canonize the Bible which have had a lot of success. However it's still far from standardized. Taking is a step further, the translations and transliterations are also a concern.
The first attempt was the Septuagint which was more of an exercise of translating Hebrew Bible and some related texts into Koine Greek. Which then became a "canon" for the Greek speaking people.
The first "Christian" canon was in 140AD and it was called "Marcion of Sinope".
The big attempts are mentioned by @crownjewl82. There are also specific "canons" for Latter Day Saints.
So, to directly answer your question, "Is there any standard version of Bible?". Assuming that the context is the entire world and everyone in it... "no".
This is more tangentially related to the question, but in a sense, there are authoritative collections.
I would like to point out that certain translations are considered definitive, though these are much more subject to linguistic evolution. For a long time, the Authorized Version (otherwise known as the King James) was the de facto standard bible translation in English (and modern versions like the NIV and the NRSV are heavily influenced by the KJV). The Vulgate serve that purpose in Latin, and the Septuagint serves that purpose in the Orthodox Church.
I think the best way to answer this question is to focus on what disputes there are about the canon of scripture between Protestants, Roman Catholics and Eastern Orthodox Christians (The Big 3).
First, a brief summary on how the translation is done (roughly)
Everything that folks said in the answers above is true: the original copies of all the books of the Bible have long been lost and we possess what has been handed down through the centuries and what is dug up by archaeologists. If you go into a bookstore and pick up a bible you will be reading a translation of what The Big 3 believe is inspired scripture. Biblical source texts are compiled from historical texts and some decisions need to be made about which text to translate from as there are slight variations between some copies from different places. A good study bible will include variant readings from different texts.
King James Only Dispute
There is an ever dwindling group of Protestant churches that believe that the King James bible (translated in the 17th century) is the authoritative Word of God for the modern world. The main argument used against later translations (that is relevant to this thread) is that they use texts discovered after the translation of the KJV which are older, but are viewed as less reliable by the KJV crowd, because they don't agree with the majority of the other sources.
Septuagint (LXX) or Hebrew Dispute
The LXX is the ancient Greek translations of the Old Testament made for Jews living in the Greek world. The Hebrew Old Testament is written in the original Hebrew. The proponents of the LXX claim that the copies of the Greek Old Testament are more reliable, because we have older copies of it and it was the text used and preserved by the first Christians; contemporary and ancient Christians even accuse the Jews of editing the Hebrew so that Jesus looked less like the Messiah (one of the Psalms says "they pierced my hands and my feet" in the LXX but "they were about my hands and feet" in the Hebrew). The proponents of using the Hebrew scriptures claim that translating from the original language is better to get the true meaning of the text (St. Jerome studied Hebrew for this reason). Again, a good study bible will include variant readings from different texts. The Orthodox church uses the LXX exclusively to translate from.
Deutero-canonical Books AKA Apocrypha
There is a collection of books that are listed as a part of the canon in the Catholic and Orthodox churches, but rejected by most Protestants (see list http://orthodoxwiki.org/Apocrypha). Even when included in the canon these books are regarded as secondary in importance. It is worthwhile to note that the apocryphal books were included in the original edition of the KJV. These books should not be confused with the heretical books that are also called apocrypha.
The Secret Gospels
A while ago someone published the "secret" Gospel of Judas. This book was never accepted by The Big 3 as scripture and is regarded as a heretical fabrication by ancient gnostic sects, but a handful of scholars made a translation of it and tried to make the case that it was just as authentic an account of Jesus as the canonical gospels.
If you really want to ask the question of how the books got to where they are, then I recommend a book about it, because its more like an adventure story than a neat logical process. FF Bruce's The Canon of Scripture was a good book. And The Lost History of Christianity by Philip Jenkins has some interesting bits about the evolution of the canon as well as dogma.
If I can summarize then I will just say that the Church decided what scripture was and who the heretics were and who the saints were. If you believe in the Holy Spirit then you will understand this process as the unfolding work of God, if you don't then you will view the church as a political body that rose to power and solidified itself by convening international councils to unify doctrine and canon.
I like to think of the process of canonization much like the building of your own family stories and traditions. The original churches received personal instruction from the apostles which they handed down and eventually shared. Once everyone had shared the letters they received from the apostles, we had the canon.