Does Paris summon the Rev. Hale to Salem?
To prove witchcraft did not exist in Salem?
To calm down the fears of the people b/c Hale was an 'expert' in demonic arts?
Paris felt that this would help him maintain/regain control over the Congregation?
Parris is terrified by the idea of witchcraft in Salem, so he sends for Hale hoping against hope that he will say there are no witches in Salem. Parris is very concerned about his position in the town. In Act I Scene I he urges Abigail to tell what she has done saying, "Do you understand that I have many enemies?" He continues to explain that "There is a faction that is sworn to drive me from my pulpit."
When he finally goes downstairs we see some of the contention as Proctor complains about his preaching of "hellfire and damnation" and Parris complains about his salary and that he is "persecuted here." Later we see more reasons that there migh be a faction rising against Parris when we hear Proctor's complaints about Parris' avarice for money for "golden candlesticks." Parris has never been comfortable in his position in Salem so when a question of witchcraft arises, he does not want to deal with such a contentious problem so he sends for Hale hoping that by bringing in the outsider, he will hold onto his position in Salem.
When asked by Putnam if he's sent for Rev. Hale, Parris replies, "He has much experience in all demonic arts,..." Parris sends for Hale because of his experience, but by doing this, he will also accomplish the other things you've listed. If Hale can prove there are no witches, fears will be calmed, and Parris will be seen as handling the situation responsibly, perhaps improving his reputation in the community. Mrs. Putnam reminds them that Hale found a witch last year, but this upsets Parris. He replies to her, "Now, Goody Ann, they only thought that were a witch, and I am certain there be no element of witchcraft here." When Putnam reacts to this statement, Parris says,
"...leap not to witchcraft. I know that you--you least of all, Thomas, would ever wish so disastrous a charge laid upon me. We cannot leap to witchcraft. They will howl me out of Salem for such corruption in my house."
At this time, Parris is scared to death for himself. The fact that his own daughter is affected scares him the most because he's afraid he'll be run out of town. Sending for Hale takes the pressure off of Parris.