When did people start to call Jesus "Christ"?

There seems to be some confusion in the question, as it presupposes a distinction that does not exist. "Christos" is the Greek translation of Hebrew "Mashiach"; both mean "anointed one". They are one title, not two. This fact resolves all these questions rather simply.

• When did people begin to call Jesus "Christ"? We see Him acknowledged as Messiah/Christ many times beginning from His birth onward (Luke 2:11,16; John 1:41; Matthew 16:16, etc.). So, as soon as people started talking about Him in Greek. Since Jesus was known to some degree among the Gentiles (see John 12:20-21), and among Greek-speaking Jews (some of His disciples had Greek names), there's no real reason to think that Greek speakers weren't calling Him "Christ" during His lifetime.
• "The priests could not have asked Him about being the "Christ" - the word "Messiah" being more appropriate as a translation" - They could have, if they were speaking Greek, but most likely, they were speaking Aramaic, so probably they did indeed say "Messiah" rather than "Christ". However, the account was recorded in Greek, and the word in the original text is "χριστὸς" - "christos". There seems little point in an English translation speculatively translating Greek to Hebrew. Most appropriate is either leaving "Christ" as written by the Biblical author, or consistently translating as "the anointed one".
• Side question: Was the name / title "Christ" used at all in the Old Testament? Yes and no. It would be very odd indeed if the OT authors, writing in Hebrew, for some reason chose to write the title of the Messiah in Greek. But in the Greek Septuagint, "Messiah" is consistently translated as "Christ".

First, in terms of biblical writing, the first use of the word "Christos" is in the letters of Paul, who wrote - by nearly all accounts - earlier than any of the gospels. Thus, Paul's readers, especially his Gentile readers, would have used this word.

Earlier, people started calling Jesus "Christ" rather than "the Messiah" when they spoke Greek rather than Hebrew. The OP is correct that the high priest would not have used the Greek term when questioning Jesus. However, Jews who spoke Greek might indeed have used the term. We have a contemporary example of this principle when American Jews speak of going to the "synagogue" (a Greek-derived term) rather than the Hebrew beit kneset (meeting house).

There is also evidence that non-Jews witnessed some of Jesus' miracles, such as his exorcisms in the Decapolis. (Mark 5:19-20) This would have stimulated discussion about his role, in which "Christ" would have been used rather than "Messiah."

In addition we have a couple of more direct clues:

John 12

20 Now there were some Greeks among those who had come up to worship at the feast. 21 They came to Philip, who was from Bethsaida in Galilee, and asked him, “Sir, we would like to see Jesus.”

Whether these men were Hellenistic Jews or literal "Greeks," they apparently spoke Greek and would have used the Greek term to refer to the Messiah. Also:

John 19

19 Pilate also had an inscription written and put on the cross. It read, “Jesus the Nazorean, the King of the Jews.” 20 Now many of the Jews read this inscription, because the place where Jesus was crucified was near the city; and it was written in Hebrew, Latin, and Greek.

Although the term "Christ" is not used here, this provides good evidence that Jesus' messianic role was discussed in both Latin and Greek, as well as Hebrew and Aramaic.

Conclusion: the original text of Mark and the other gospels was Greek, so the word "Christ" is correct. People who spoke Greek would have called Jesus "Christ" rather than "Messiah" during his lifetime. Paul is the first NT author to use the term in writing.

Side question's answer: the word "Christ" was used in the Septuagint version of the OT, which was in wide circulation in the Roman empire and was used by most of the NT writings when they quoted scripture.

STRONGS NT 5547: χριστός

χριστός, χριστη, χριστόν (χρίω), the Sept. for מָשִׁיחַ, anointed: ὁ ἱερεύς ὁ χριστός, Leviticus 4:5; Leviticus 6:22; οἱ χριστοι ἱερεῖς, 2 Macc. 1:10; the patriarchs are called, substantively, οἱ χριστοι Θεοῦ, Psalm 104:15 (); the singular ὁ χριστός τοῦ κυρίου (יְהוָה מְשִׁיחַ) king of Israel (see χρῖσμα), as 1 Samuel 2:10, 35; (1 Samuel 24:11; 1 Samuel 26:9, 11, 23); 2 Samuel 1:14; Psalm 2:2; Psalm 17:51 (); Habakkuk 3:13; (2 Chronicles 22:7); also of a foreign king, Cyrus, as sent of God, Isaiah 45:1

First, the translation in Mark 14:61 is accurate. We find other such examples from the very earliest times. Christ (Greek) and Messiah (Hebrew) both mean anointed one, a title of Jesus.

• Matt 1:16 - and Jacob the father of Joseph, the husband of Mary, of whom was born Jesus, who is called Christ.
• Matt 1:17 - So all the generations from Abraham to David were fourteen generations, and from David to the deportation to Babylon fourteen generations, and from the deportation to Babylon to the Christ fourteen generations.
• Matt 2:14 - And when he had assembled all the chief priests and scribes of the people, he asked them where the Christ was to be born.
• Matt 11:2 - Meanwhile John heard in prison about the works of Christ, and he sent his disciples
• See also Matt 16:16, 20, 22:42, and many more.

Thus, Christ is a title, not a name. Jesus was His name and sometimes He was called simple, Jesus, or Christ or sometimes, Jesus Christ, and sometimes, Christ Jesus, etc.