No he shouldn't be doing it, Gracesgran. I think they do it because they can. Social media has made it that much easier to get your message across to so many people at far less expense than sending out leaflets etc.
If I were Theresa May, I'd give the Three Merry Brexiteers a bit more rope and hope they hang themselves (metaphorically). Boris stood down from the leadership candidacy and seems to delight in meddling. Fox and Davis are also unsuccessful leadershop contenders. Tough luck if they don't like what May's telling them to do. She's the boss.
And to answer your question about who to make International Trade Secretary, I'd appoint Jeremy Hunt - anything to get him away from Health . I believe he speaks Mandarin and, to be fair to him, he has set up some successful businesses, unlike Fox.