I'm not quite sure what you mean by "original papers" or "first press," but here's a bit of the textual history.

Socrates wrote in the 5th century, but the earliest copies are from the 11th or maybe the late 10th century--this is not unusual (sorry--that's an important distinction, oops). Both medieval manuscripts live in the same library in Florence, the Biblioteca Medicea Laurenziana. They are known as MS Plut. 69,5 and MS Plut. 70,7. In between Socrates' life and those early Greek manuscripts, the text was known to and excerpted/translated into Latin, Armenian and Syriac.

The first printed edition, in Greek, was published by Robert Estienne in 1544 Paris based off a single newly-copied manuscript (itself based on a copy from the previous century...having fun yet?).

The most important early edition was published in 1688 by Henry de Valois, also in Paris. Valois consulted multiple manuscripts when assembling--"editing" is the scholarly term--his edition, in the hopes of yielding a version closer to the original (scholars have developed ways of "reading backwards" through manuscript traditions).

In Greek, there are several "modern" printed editions floating around. Most of them are actually 19th or even 18th century translations republished in the 20th, including the Patrologia Graeca version.

The current authoritative Greek edition is Gunther Hansen's Sokrates: Kirchengeschicte for the Griecheschen Christlichen Schriftsteller series (1995).

The most common English translation you will find online is from the Nicene and Post-Nicene Fathers series, published by A.C. Zenos in 1890, although there are some earlier ones on archive.org (thus elsewhere, one presumes).

I hope this helps!