If 100% of your compensation is a properly designated housing allowance (meaning it was designated in advance and in writing by the board) then you don't get any tax forms. They should just issue you a letter stating the amount of housing allowance for your records.
You will need turbotax self-employed (online) or Deluxe or higher (CD or download). Enter that you have a small business. Use code 813000, that will bring up a question to enter your housing allowance. If you have cash side income such as honoraria for weddings and funerals, enter that as additional cash income not on a 1099-MISC.
You will not be allowed to deduct any work-related expenses such as mileage or supplies that you paid for since all your compensation is income-tax exempt (called the Deason rule).
If the housing allowance was not designated in advance and in writing then it's taxable compensation and you should receive a W-2. Although clergy are treated as self employed for some tax purposes, if you answer to the church (via direct election or a board) or if you answer to a denomination, you are a common law employee and should have a W-2 (except in the case when 100% of your compensation is a housing allowance.)
You and the church probably need this http://www.ecfa.org/PDF/2016-Preparing-Tax-Returns-For-Clergy.pdf
So does the church notify the IRS of the income they have paid me at all? I'm asking these questions for our Treasurer who has never had to deal with this situation before.
Also, according to the Deason rule, while I cannot deduct expenses incurred for the church, I can still deduct expenses for my other business -- correct?
If 100% of your comp is housing allowance, then the church does not need to send anything to the IRS, either W-2/W-3, form 941 or 944, or 1099-MISC (unless they have other employees, of course).
If you have a second self-employed business then you need two schedule C's, one for each. You can deduct expenses for business #2 on schedule C #2 as usual. You just can't deduct pastor expenses on your pastor schedule C. (In fact, I'm not even sure if you will have a schedule C if you don't have additional cash income to report, you may just have a schedule SE.)
And of course, even though the housing allowance is excluded from income tax it is still subject to SE tax, so your schedule SE should include both the housing allowance and the net profit from business #2.