Hi,
Both are OK. Present tense is OK as long as the fact is true, or believed to be true, today, and the 'telling' was fairly recent.
1.He told me that there IS a city called New York.
OR
He told me that there WAS a city called New York.
2.He told me that the Earth is rotating.
OR
He told me that the Earth was rotating.
3.She thinks that she is pretty.
OR
She thinks she is pretty.
(Is it always necessary to put "that" in sentences like this?) No.
Best wishes, Clive