If I have a mastectomy, does that mean I won't need other treatment?

Not necessarily. Breast conservation and mastectomy are the 2 primary local therapy options. While for treatment of noninvasive disease (dcis) this may suffice, w/invasive disease systemic therapy must also be considered to address microscopic disease elsewhere that can't be reached w/surgery or xrt. More advanced (ie. Large tumors, multiple nodes +) may warrant post mastectomy XRT to maximize local control.
Not always. Even after mastectomy we recommend some sort of adjuvant treatment for most patients, typically hormonal modification (tamoxifen, arimidex, (anastrozole) etc. And or chemotherapy. These drugs are given to patients at risk for metastatic disease that is too small to be picked up with scans, etc. Radiation to the chest wall is recommended after mastectomy for large tumors, those involving the skin, or lymph nodes.
Possibly. You may need additional surgery such as removal of your lymph nodes, radiation and/or chemotherapy, or an operation on the opposite breast. You may choose reconstruction, which may be anywhere from 1 or more additional surgeries. Make sure you understand the process before undergoing any surgery.