How high are the chances of having breast cancer and cervical cancer at the age of 24?

Mixed picture. Outside of genetic causes, breast cancer in a 24 year old is very uncommon, although it can occur. Cervical cancer, on the other hand is not uncommon in that age group, and would be much more of a concern. Fortunately, having regular pap smears is an excellent way of screening for the precursors of cervical cancer and should be part of your healthcare plan.
Very low. For most women, so low as to be nearly zero. However, there are familial cancers that affect the breast and ovaries (rather than cervix/uterus), and in those families cancer can strike in the 20s.
Not THAT low... Cancer of the cervix, or at least precancer picked up on a pap smear, is actually quite common in women in their 20's. And breast cancer, especially in its preinvasive form, isn't a rarity either. It may be dumb luck, or a cancer family syndrome such as li-fraumeni may be operating. Thankfully, the large majority of both types of cancer are cured, especially if caught early.

Related Questions

My chances of having both breast cancer and cervical cancer at the same time?

Low. While just about anything is possible, and while breast cancer & cervical cancer are not related to each other, it would be rare but not impossible to have 2 unrelated cancers at the same tome.
Low, but... Both cancers are common in women. Symptoms or signs of one do not make it any less likely that you have the other. Both are generally treatable especially if caught early; be sure you allow both concerns to be addressed.

At what age should you be screened for breast cancer or cervical cancer if grandmother and aunt had each at early age in 30s/40s? Brca testing? ;

Screening. The pap smear is the standard screening test for cercival dysplasia/cancer. You should have routine pap smears starting when you first became sexually active. Mammograms are effective screening tools especially as women age, they are not effective for younger women with dense breasts. Speak with a genetic councilor about brca testing. Sounds to me like you should be tested.
Risk assessment. Cervical cancer is mostly related to hpv virus, you get your paps as any other woman, starting around 21. You should talk to a genetic counselor to get a good family history and probably get tested for brca and other high risk genes. Even if negative, and depending on the circumstances, you may be considered high risk and be a candidate for enhanced screening or risk reduction manipulations.
Different for each. You've had sex, so you should have yearly pap smears to detect cervical cancer. (this is not related to brca.) if your aunt and grandmother are on the same side of your family, and they both had breast cancer, you may meet criteria for brca testing. Genetic counseling is the next step. You don't need mammograms until 40 (unless you have a mutation), but do breast self-exam monthly!
See genetics. Early age breast cancer might signal an inherited gene mutation such as the BRCA1 or BRCA2 gene, for which screening would begin around the age of 25. See a geneticist to assess risk for an inherited cancer syndrome. Cervical cancer screening begins by age 21. Family history may be a sign of shared environmental or lifestyle factors but inherited risk unlikely; tell your GYN the family history.

Is there any links between cervical cancer and breast cancer?

Ovarian cancer. And breast cancer are related in that certain genetic defects make a person more susceptible to develop both. No such correlation with cervical cancer. Causes of malignancy are multifactorial, and we are just beginning to understand them.

Would metastatic cervical cancer in the breast appear the same as breast cancer on mammogram and ultrasound?

Possibly on imaging. Cervical cancer (the tissue at the end of the vaginal canal) rarely metastasizes to the breast. Rather breast cancer may metastasize to many areas of body, such as bone, lung, brain, lymph nodes. That being said, any abnormal growth in the breast, though mostly breast tissue, could look similarly by mammogram or ultrasound. The bottom line is if there is an abnormality, must be biopsied.
Perhaps. While it would be highly unlikely for cervix cancer to spread to the breast, if it did, plain mammography would likely not differentiate the two processes.
Hmmm. It would be unlikely that cervical cancer would metastasize to the breast. While rare, metastasis to the breast have slightly different characteristics on mammogram and ultrasound from primary breast cancers, but they can look similar enough that a biopsy would be needed.

Does getting your first period at an age lower than 13 actually increase your risk of ovarian/cervical/breast cancer?

Minimal effect. Statistically speaking it may slightly increase the risk of breast cancer and Ovarian cancer. But in an individual person it makes no big difference, so I would not worry about it as there is a long life to make amends and live by a healthy lifestyle and overcome any such risks. If you eat healthy and exercise daily, the risk of cancer goes down considerably!

Is there a correlation between cervical dysplasia and breast cancer?

No. Cervical dysplasia is associated with the human papilloma virus (hpv). As far as we know, breast cancer is not associated with hpv or other viruses.
No. There is not correlation. Breast cancer is often associated with ovarian cancer especially in brca mutation carriers.