Psychological pre-hire screening is done at most all major US carriers. These are generally pencil and paper tests. Some are quite involved like the MMPI, Minnesota Multi-Phasic Inventory. I think this test has over 1000 questions or statements with which the candidate is supposed to answer as agree or disagree. Delta used to have an interview with a psychologist.
I don't think any US carrier has testing after hiring. ALPA does have several programs for intervention in stressful situations. HIMS, is a well known peer operated intervention program for substance abuse. American has project Wingman where volunteers offer assistance in getting support for problems with kids, divorce, substance abuse, and other situations.
Overall the FAA and the Companies have been supportive of these programs as they offer good professional counseling and support but are administered and run by Pilot Unions. Pilots are more prone to trust their peers and a peer run support network rather than the company or the FAA.