For the most part, this isn't a history question at all, but a linguistic one! Almost all of the names you listed are what language scholars call "diminutive." Diminutive words are usually a root word with something added to it (usually a suffix, sometimes a prefix) to indicate that it's smaller or lesser. Given the English language's historical penchant for incorporating words from other languages we use quite a lot of them. Other English examples include -let (as in drop > droplet), -le (spark to sparkle), -ling (duck > duckling), and -ish (tall > tallish).

-y/ie/ey does have a bit of historical explanation to it. Broadly speaking it is related to a whole series of suffixes in many Indo-European languages, especially other Germanic languages. It's even related to another -y suffix in English: when -y is used to make a verb or noun into an adjective (as in run > runny or mess > messy). When applied to a noun in the diminutive sense -y is what linguists call "assimilative." That means we use the diminutive effect of implying smallness to indicate a more personal and less formal relationship. A non-name example would be mom > mommy.

In concept, thus assimilative form of -y is something common to the western North Sea Germanic languages that have rise to English, Scots, Dutch, and a few other languages but first became popular in Scots and Dutch. Interaction with those two languages reinforced it in late Middle/early modern English. The 1400-1500s mark the beginning of common diminutive names like the ones you listed.

Middle English had several of its own common ways of forming nicknames or diminutive names and sometimes these mixed together. One common option was to take the first syllable of a name and replace the initial letter/sound. This is related to modern Cockney slang, but was largely abandoned as a source for nicknames except in isolated cases. One of those is Peg as a nickname for Margaret (Marg > Peg in Middle English). Once the -y diminutive became common Peg could become Peggy, and Peggy became a diminutive nickname for Margaret.

The lone An* example on your list that doesn't fit into this diminutive schema is "Jeff/Jeffrey." Jeff is actually the shortened (and therefore diminutive) form in this case.

Jeffrey is just a variant spelling of Geoffrey. Geoffrey in turn a slightly altered form of the Germanic name "Godfrey" or "Gottfried." Gottfried is a compound of Germanic fred, meaning "peace" and either "God" or gaut, a word related to Goth (as in Visigoth). As a name it goes back to at least the 8th Century.