I am writing a story set in the 15th Century. I appreciate that most stories etc were written at the time in either Latin or French but, for obvious reasons, I have to write it in English. It is written in the 1st person and, to make it more personal and immediate, I am considering using abbreviations such as 'I've' and 'I'm'. Would such abbreviations have been used in 15th century English?
Middle English at the time did have a contraction for "I have" (and I plus other verbs), but it's not the same as "I've": ichabbe, ichave. These forms reflect the pronunciation of ich, which was the form of I inherited from Old English and was undergoing a shift in pronunciation to its modern pronunciation and spelling.
I don't see any evidence of similar contractions being used with other pronouns during the 15th century.
The only modern negative contraction that I can think of that was in use in the 15th century is cannot. From a c1450 Treatise on Fishing:
Ye cannot brynge a hoke to a fyche mouthe but yf þer be mete ther-on to hys plesur. (You cannot bring a hook to a fish's mouth unless there is meat on it to its satisfaction.)
There are other words like it (such as wilnot) but none of them survived to modern English. Middle English also still had the negative contractions from Old English (ne + is = nis), though these are even harder for a modern audience to decipher.
As for when people started using modern negative contractions: "The contracted forms [of not] seem to have come into speech ... about the year 1600" — Negation in English and Other Languages in The Contractions of not: A Historical Note (which explains in more detail about how n't came to be).
There are some other contractions that existed at this point, but they're not terribly useful in the modern age.