in the sentence: "The requirements of the legal system must be followed in order to protect the reliability of evidence and thereby facilitate THEIR introduction to court." do we use the plural pronoun THEIR or singular ITS?
Yes, in this case it should say "its" instead of "their". When discussuing evidence or introducing it in court it is usually referred to in individual terms , as in "several pieces of evidence were found at the crime scene" or "Your Honor, I would like to have this gun marked 'Exhibit A' and entered into evidence."
"The evidence is overwhelming" is a frequently used phrase, so it follows that "evidence" on its own is treated as singular.