How do you prove to non-members that Joseph Smith was a prophet of God? What sins need to be confessed to a bishop?
In answer to your first question, you don’t! An intellectual proof of the authenticity of the prophetic calling of the Prophet Joseph Smith is the easiest thing in the world to prove. The evidence is absolutely overwhelming! But since it touches people’s lives and the acceptance of that proof implies a moral obligation on the part of those who accept it, it is generally denied. The only valid proof of the prophetic calling of Joseph Smith and of the kingdom that he restored is obtained by the witness of the Holy Spirit. If the Holy Ghost tells you it is true, you will believe it! Can anyone imagine that the Holy Ghost is a poor communicator? So the thing to do, is to convince your non-member friends to ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost (Moroni 10:4).
You might have them listen to the discussions with the missionaries so they will know what to ask.
Concerning what sins need to be confessed to your bishop, I would suggest all sins of a serious nature, and especially those that you might need some help with in the repentance process.