Does a MRI always show a tear? What physical exam test is giving for ACL problems?

No. Mri almost always shows meniscal or acl tears but nothing is perfect.
Drawer test is used. To test for tears in the cruciate ligament(s) of the knee. It may not detect slight tears, whereas an MRI can show even small tears. A good orthopedic surgeon can test you to determine whether an MRI is necessary.
Not always. But an MRI is about 96 % accurate in detecting an acl tear. A lachman physical exam and anterior drawer physical exam can be used to diagnose an acl tear.