How many New Testament manuscripts do we really have? How does the Bible size up with other books from the ancient world?

Is it really true that the New and Old Testament is the book from the ancient world that has the largest amounmt of extant manuscripts?

Into how many ancient languages has the Bible been translated?

And finally, if we did not have all the manuscript evidence some believe we could still get the gist of the New Testament from early church correspondence. If this is true how does this improve or maintain the general case for an historically accurate New Testament?
Old Testament

The Dead Sea Scrolls currently housed in the Israeli Dead Sea Scroll museum and available online, is a collection of 972 manuscripts from the period 408 - 308 BC.  Seeing as the Old Testament itself consists of books written between 1000 - 600 BC and the late 450s, this is a pretty good collection. There are others, but this is the most well-known and complete collection.

New Testament

According to the site Christian Apologetics Research Ministry, there are currently 5,686 known fragments of New Testament manuscripts.  That site also shows how much closer to the date of composition these manuscripts are than many other contemporary texts. The earliest of these (p52) dates to sometime between 96 and 125 AD.  Seeing that some people date John to the 90s (others to the 60s), the gap is extremely small indeed.

More interestingly, a good apparatus (like this) can actually show you every variation on the texts, and you will see they are few and far in between. This question shows how an apparatus works.

Other Questions

Finally, you said that we could probably piece together the gist of the bible from early church correspondance.  This is completely accurate - as many of the early church fathers were already quoting New Testament Scripture by 100 AD  (Martyrdom of Polycarp? 1 Clement? the Didache!)  Indeed, Contra Celsius, in 150 is sometimes used to verify reconstructions.
Even the existence of extra-biblical works, both pro and con lend credence to the theory that at least "something" was already going on about Jesus pretty shortly after the events of the New Testament.

More importantly, however, is the fact that much of the New Testament actually is correspondance between churches.  Epistles are just that - letters (often from Paul) to churches that predate the Gospels and give a "second witness" to the Good News.