“In Mark 6:7, Christ sends the apostles out to preach two by two. Was Judas involved? If so, with whom was he paired?”
If the Lord didn’t send Judas out to preach, the other apostles would have been suspicious of him. That suspicion would certainly have manifested itself at the last supper, when the Lord said, “One of you which eateth with Me shall betray Me” (Mark 14:18). If they had any reservations about Judas at all, they would have asked, “Is it Judas?” Instead, “they began… to say unto Him one by one, Is it I? and another said, Is it I?” (v. 19), “doubting of whom He spake” (John 13:22).
So it seems clear that Judas preached the gospel. If he believed the gospel, he must have believed it in the shallow sense in which others in Scripture are said to have believed (John 2:23,24; 8:30 cf. 31-44; Acts 8:13-23). That is, he didn’t “believe to the saving of the soul” (Heb. 10:39).
We’re not told who he was paired with, but if the apostles got to choose, you would think he’d have chosen Doubting Thomas. If he was already showing signs of the doubt he expressed later (John 20:25), then Judas would no doubt have felt most comfortable with him.