Does an ultrasound of the breast mean I don't have to take a mammogram?

No. Ultrasound is used to further define mammographic findings. It is not a substitute for the mammogram.
No. As a diagnostic tool, an ultrasound usually supplements rather than replaces a mammogram. However, if you are pregnant, only rather extraordinary circumstances would justify the exposure to x-rays entailed in a mammogram, so an ultrasound may be the first choice.
No. A breast ultrasound is usually done to evaluate a specific finding identified on physical exam or on a screening mammogram. It does not count as an overall screening test for breast cancer. An annual mammogram still needs to be performed.
No. A screening mammogram along with a doctor's examination are the current recommended first line screening modalities. A mammogram may have up to a 15% false negative rate especially in women with dense breasts. A breast ultrasound may be performed if a lump is felt or a suspicious area is found on ultrasound. Sometimes a needle biopsy may be indicated. Sometimes a breast MRI is done.
No. It really depends on your age, your risk factors and why you got the ultrasound in the first place. Ultrasounds are used to evaluate a breast mass that you can feel. They will not pick up microscopic breast cancer. Mammograms will pick up microscopic breast cancers (before you can feel it).
No. The ultrasound and mammogram complement each other, but neither is fail proof on its own. The mammogram is much better at picking up small microcalcifications or finding lumps. It is the best tool for screening. The us is good at characterizing lumps or differentiating cysts from solid lumps. The us is most helpful is something is felt on exam or seen on the mammogram.
No, they. are complementary tests, not mutually exclusive. Sometimes one shows an abnormality not seen on the other. Mammography is still standard of care for screening. If you had a screening ultrasound, it would still be valuable to have a screening mammo, even if you have dense breasts.