How should he be played on stage? What do we learn about his relationship with Faustus?
As an otherworldly being, I would expect Mephistopheles to have a detached air about him, the way he speaks is often short and snappy "Arch-regent and commander of all spirits.” like a soldier responding to his captain, this is rightly so as M is a servant of Lucifer.
Though M is Faustus' servant and companion throughout the play, we must remember the true reason he is there - to secure Faustus' soul. The first line spoken to M by Faustus is "I charge thee to return and change thy shape." Faustus chooses to be ignorant of whom and what M really is: a devil. Therefore M in my opinion is always running a double act: to happily fulfil Faustus' desires and be his friend "Nay I know not, we shall be cursed with bell, book, and candle (joking with Faustus)." whilst bringing a firm hand when he falters from the pact "Revolt, or I'll in piecemeal tear thy flesh." "Think thou on hell, Faustus, for thou art damned."
Simultaneously though M for the most part speaks with a subordinate tone to Faustus after the pact is signed and done. Before that, in Act 1 Scene 3 M is less respectful to Faustus taking on a more teacher role to Faustus in lines 78-84 when answering questions on Hell. This surfaces the idea that M is manipulative, acting according to whether he sees the potential to gain a soul for Lucifer.
M does not seem to care much for hold his relationship with Faustus if we carry on the idea he is only working for his soul. As we can see in Act 3 Scene 2, he appears to Robin and Rafe in hopes of securing more souls, we can see that this is a regular routine for M and Faustus is just another soul on the list "(Aside) O, what will not I do to obtain his soul?" - Act 2 Scene 1; M is just a desperate devil out for a desperate soul.
M is not without a sense of humour and takes his own delight in playing tricks on Faustus bringing Faustus a devil when he asks for a wife "Tell, Faustus, how dost thou like thy wife?" and if we interpret M to be playing dumb with the Horse-Courser "Come, villain, to the constable." Faustus' leg falling off may have been M's prank.
M's relationship with Faustus sometimes takes on a parental bond. Act 2 Scene 3 like a parent tired of a child's pestering questions, M ceases to answer Faustus' questions when they reach the subject of who created the world. Act 3 Scene 1 M refers to Faustus with the endearment 'my' in the line "And now, my Faustus, that thou mayst perceive", as if he has brought a child to a play area (Rome). Act 4 Scene 1 M watches over Faustus as he sleeps and would rather he not be disturbed "I tell thee he has not slept this eight nights." M sympathises with Faustus' tiredness.
Nonetheless whatever kindness, friendship and amusement M provides all fall back on his desire for Faustus' soul in Act 5 Scene 1. This is shown by M's anger at Faustus' near-repentance and desperation to rewrite the deed "Do it then quickly, with unfeignéd heart," M's threats "Lest greater danger do attend thy drift." and M's re-assumed 'kind' persona "thou shalt desire Shall be performed in the twinkling of an eye.".
Faustus on the other hand seems oblivious to M's manipulation, or perhaps too desperate for a companion to convince himself M is only out for his soul. Faustus' final line "Ah Mephistopheles!" may be spoken in sad relief that his fate is to be with his one and only friend or anger at the damnation M has brought on him, either of these, it demonstrates the huge role M has had in Faustus' life.