Your first point: A scar visible on x-ray of the chest after complete, supervised tuberculosis treatment
is not uncommon. Let us presume that your mom received a full (at least 6 months) supervised treatment. Usually sputum is tested at the end of treatment to confirm that there are no tubercle bacilli.
Your second point: Yes, a repeat sputum study for tubercle bacillus is in order, from a reputed laboratory (preferably a laboratory under the local government, accredited by the World Health Organization). This will confirm that she is not dangerous to others and is not likely to spread the disease. It will help her at immigration, if she produces negative sputum results. The sputum study can be done in 2/3 days' time. Please remember to get two samples of early morning sputum (not saliva) tested and the results signed by a doctor.
It will be even better to carry a negative tuberculosis culture report. That too is possible in a few weeks' time. If mom's old medical treatment documents are traceable, they would be of help for clearance at immigration.
Wishing your mom the best of health and a pleasant stay in the USA,