Is there any regular screening done for cleft lips?

Not usually. The usual ultrasound studies are to verify fetal growth and gestation & will not generally provide that info. If you have a cleft in your family and desire the info, a higher definition ultrasound could be scheduled. I know of no blood test or other type of screening that would be applicable.
Cleft lip screening. Yes, routine prenatal ultrasonography can identify a cleft lip (but not a cleft palate) as long as the baby cooperates by not covering its face.
No. Not routinely unless the ultrasound is of high resolution and the area in question is visible to the technician.
Yes. Prenatal ultrasound is done routinely. And if of high quality often can pick up cleft lip defects.