The actual dates are not known. However, she was probably born in the middle of the first century B.C. and probably died early in the first century A.D.
in judea
His grandmother
He lived in the desert.
John the Baptist's early life could have been in a monastery with Essene monks.
According to Luke 1:38-40, Elizabeth lived in the 'hill country of Judea.' That is where Mary mother of Jesus went to visit her.
The Bible is silent in regards to your question. However, Elizabeth and Zachariah were well along in years when the angel Gabriel appeared to Zechariah in the Holy of the temple and announced that Elizabeth would give birth to a son who was to be called John. (Luke 1:5-7, 11-13, 24)
John the Baptist lived in the wilderness for an unspecified amount of time before beginning his ministry.
John the Baptist chose to live in the wilderness to separate himself from worldly distractions, focus on his spiritual calling, and prepare himself for his role as a prophet and forerunner of Jesus Christ.
Live in the desert, survive on bugs and honey, preach Jesus' coming and redemption through baptism. IE your not John the Baptist, nor will you or anyone else ever be.
Went off with their mother Harriet to live.
John the Baptist was not a Christian, as we typically think of Christians. John lived and died before Christ's death and resurrection, so he had no opportunity to become a Christian, per se. However, John the Baptist clearly lived a life exclusively dedicated to obedience to God, which is obviously a virtue Christians are commanded to exhibit.