The Governor of New Jersey is authorized to make an appointment to vacancies in the United States Senate. I personally propose he do so and set a special election to take place in June 2016.
No. As New Jersey elects its state legislature on odd years, if the seat is vacated before April 3rd, the election will have to held in November 2015 (or before), according to one reading of the law. The other reading is saying than it would be if it vacated before early September, it would have to be in November 2015.
You see, New Jersey has general elections, this year.