The actor is accused of sexual assault or harassment by 13 women, a French news website reports.
The interview covered everything from the challenges of artificial intelligence to Twitter's current CEO.
The exhibition will feature garments worn by the likes of Whitney Houston and Sir Elton John.
The co-host says he will leave It Takes Two after four years to "explore what else life has for me".
The Twitter boss changes the BBC's description from "government funded media" to "publicly-funded".
Musk talks about the "pain" of owning Twitter and the BBC's objection to being labelled "government funded".
The multi-billionaire defends his running of the firm in a hastily-arranged interview from Twitter HQ.
The Angel of the North is another UK landmark to feature in the competition's introductory clips.
In a last-minute interview with the BBC's James Clayton, Mr Musk was asked about the "government funded" Twitter label.
A judge has ruled the start-up founder could not stay free while she appeals against her convictions.