As for the reason Trump would do this - Tyler Cowan's Bloomberg column is getting a lot of attention, and deservedly so:
By requiring subordinates to speak untruths, a leader can undercut their independent standing, including their standing with the public, with the media and with other members of the administration. That makes those individuals grow more dependent on the leader and less likely to mount independent rebellions against the structure of command. Promoting such chains of lies is a classic tactic when a leader distrusts his subordinates and expects to continue to distrust them in the future.But read the whole thing...
Another reason for promoting lying is what economists sometimes call loyalty filters. If you want to ascertain if someone is truly loyal to you, ask them to do something outrageous or stupid. If they balk, then you know right away they aren’t fully with you. That too is a sign of incipient mistrust within the ruling clique, and it is part of the same worldview that leads Trump to rely so heavily on family members.
In this view, loyalty tests are especially frequent for new hires and at the beginning of new regimes, when the least is known about the propensities of subordinates. You don’t have to view President Trump as necessarily making a lot of complicated calculations, rather he may simply be replicating tactics that he found useful in his earlier business and media careers.