To your first question, I shall reply - yes, there appears a great conflict in in this play, which is one of the greatest actually in literary history. This conflict is between Faustus with his own-self. But conflict between good and evil also is a significant conflict providing the plot a grand theme.
Faustus is a prominent scholar of Germany having huge competency in almost each arena of knowledge. He is now willing to know more, his curious mind drives him to explore newer things through which he would be able to gain limitless power. Soon, he gets attracted to necromancy which, he thinks, would serve his purpose. This black magic compels him to go through a pact that he must sell his soul to the gang of Lucifer if he wants power via necromancy. As he believes that a "sound magician is a demi-god", he deals with them. But, the conflict arises from here which is with himself. He finds out an alarming sign on his hand which says him "Homo Fuge" or to fly, meaning not to sign the deal, in fact his speech "be resolute:/Why waver'st thou?" expresses clear dilemma. But he gets defeated by the evil power and by his greed. He, many a times, thinks of abandoning the devilish path. He says to Mephostophilis at one point: "I will renounce this magic and repent". He, in the scene where he is threatened by Lucifer's gang, calls Christ and curses Mephostophilis, but gets scared by the devils later. The old man's speech clearly defines that the "amiable soul" is getting contaminated, and Faustus seems to realise his sin, yet, after a while, Mephostophilis distracts him using his passionate lechery and love towards classical beauty. Even, in the finishing part, he wants to repent, though, he can not. Thus, in the entire play, we see, the protagonist is fairly stuck in a dilemma or a conflict between his two selves - one which knows that what he is doing is heinous, and the other which considers glory, fame and immortality equal to heaven.
Inside the above sort of conflict, another conflict is hidden - conflict between Good and bad. The good angel, the old man, even Faustus himself - all act as good characters which fight against the evil - Lucifer, Mephostophilis, Beelzebub, bad angel and more importantly Faustus the magician. The play itself is a morality play which has a everyman - Dr. Faustus, and depicts combat between good and evil, and lastly through showing the defeat of evil, provides a moral. The right wins at the end.
Now, come to the second question. Mephostophilis is a significant character in Doctor Faustus.
As I have already discussed, he is definitely a malevolent character, yet, he has a bit dimension in his characteristics. He is such a slave of Lucifer who is trapped in the circle of hellish torment. His own utterances - "Think'st thou that I, who saw the face of God/ And tasted the joys of heaven, / Am not tormrnted with ten thousand hells/ In being deprived of everlasting bliss?" - convey his anguisg and pathos which, for a while, evokes a bit sympathy towards him.
Still, The malevolence of Mephostophilis undoubtedly overshadows all these. He is a villainous catalyst. His functions are to provoke the devil inside Faustus and to lure him.
Yet, it is important to note that Mephostophilis is just an instigator, the main control over the fate of Faustus is in the hands of Faustus himself. For the tragic consequence, Faustus himself is responsible, Mephostiphilis is solely a side-part.