What are the survival rates for the various types of breast cancer surgeries? Does the type of surgery affect whether the cancer will reappear?

Equal survival. The survival rates for breast conservation and mastectomy are the same. There is a slightly higher rate of local recurrence for breast conservation (with radiation it should be <1% per year out to 10 years) but this does not impact survival which is the most important consideration of any cancer treatment.
The same. It is well-established that the survival rate for breast cancer is the same whether a mastectomy or lumpectomy is performed. However, breast cancer can "come back" in the treated breast more commonly following lumpectomy. This is why radiation therapy is necessary after lumpectomy, which will bring the local recurrence rate down below 5%.