On the Healing of the Paralytic
I just finished reading about a woman who, in the wake of a series of illness-related deaths in her family, developed her own case of incurable cancer - without ever discussing her own reaction to the family tragedy. In order to help her prepare for her coming death, she started taking therapy and finally discussed what had happened to her sisters. Her cancer then went into full and complete remission and has not since returned. (it's from Edwin Friedman's From Generation to Generation).
Now that's not biblical exegesis per se, but if the Orthodox have been right for the last thousand years that sin is an illness in a very real sense of the term, reading the healing of the paralytic isn't all that difficult to understand. Sin is a cancer, sin is paralysis. The point is the forgiveness and the liberation. Forgiving heals. This is why the authority for forgiveness matters, particularly to Matthew: it is a very great power. Perhaps that is why Luke, in its focus on spiritual powers, jumps to the Pharisees so quickly - they don't, after all, even attempt to heal the man themselves. They can't. But Jesus can, says Matthew, and can because he has been given authority by God.
If indeed the Matthean community was an offshoot of Temple Judaism, there would indeed be the question of authority. That's probably the first question that comes up in 'alternative communities' - where does the authority to do these things come from? In this case, it comes from Jesus and we know that from stories like this one. In Jesus' case, it came from God.
The other differences between the gospel are perhaps less overtly significant. Luke doesn't place Jesus anywhere with this story, but says that the people had come from everywhere - from all Judea and Galille and Jerusalem - owing to Luke's theme of universal salvation. Only Matthew has Jesus, in response to the accusation of blasphemy, ask the Pharisees why they hold evil in their hearts - because Matthew is increasingly defining that community over against the Temple Judaism the Pharisees represent.