I am sure the original Jacques had only one wife, but as you have found, her name is sometimes quoted as Jacqueline, sometimes Catherine or even Marie Catherine. It appears to me that French language sites tend towards Jacqueline, English ones Catherine. I don’t know whether they could be considered translations.
As for her surname, it would have been either Pin or Pain, which are pronounced identically in French. I presume that the version Piri comes from misreading old gothic “black-letter” script, where a dot over an “i” can be indistinct, and a dirty mark over the second downstroke of the “n” can make it look like “ri”. The version Pire or even Piré (with an accent) I suspect comes from someone seeing “Piri” and trying to make it look more French.
Yes, “before December 31, 1664” undoubtedly refers to their son being born in that year. I have seen a date of 1651 for their marriage, but unfortunately I haven’t recorded the source of this.