Is sonomammogram able to know if you have cancerous cyst?

No. Sonogram can distinguish between solid and cystic mass. It can also give you information about echogenicity, septation and boarders which sometime help characteristize the mass and describe it as highly suspicious for cancer or likely benign. A good pathology specimen is the only confirmatory test.
Cyst vs solid. Mammography can detect breast masses with 90% accuracy. Ultrasound can distinguish if the mass is solid or fluid-filled. Simple cysts, those that are smooth and filled with clear liquid are 99% benign. If a mass has cystic features, the other features may characterize it as benign appearing or suspicious. When in doubt, biopsy is indicated.
No, a. biopsy would be necessary to make that determination. However, based on various ultrasound features, the radiologist can classify the lesion with respect to the likelihood of malignancy. This is called the BIRADS classification. A cystic lesion could be classified, in order of increasing suspicion, from BIRADS 2(benign) through 5(>95% chance of malignancy).
Not alone. These studies can characterize the likelihood of something being benign or cancerous, however, only a biopsy can determine if something suspicious but indeterminant on imaging is truly cancerous or not.