Can the doctor detect birth defects at 3 months?

Expert ultrasound. An ultrasound examination might show some abnormalities, but many might not be visible at 3 months. Also not all birth defects (congenital anomalies) are associated with chromosome abnormalities so screening the blood of the pregnant woman or testing the placental tissue would not detect them. Testing of the fetal blood or amniotic fluid isnt typically done early may be referred to experienced mfm.
Yes, in more than on. Ultrasound and maternal blood tests (aka genetic screen) can detect risk of birth defects by the end of the first trimester (10-14 weeks from the first day of last menstrual period.) if a birth defect is suspected, other diagnostic tests such as chorionic villi sampling and amniocentesis as well as repeat ultrasound can help to diagnose the specific problem.
Yes. Depending on why one would be lookng, a high resolution, high level ultrasound and a biopsy of tissue in the womb (chorionic villus sampling) along with blood testing can pick up birth defects. There is much more to consider. But yes it can be done.
Perhaps. Down syndrome can now be detect by blood test by 12 weeks. Most fetal anatomical problems do not manifest untl 16-20 weeks, although some major defects can be gleaned as early as 12-14 weeks in expert centers with transvaginal prenatal sonography.