Do periodontal ligament problems just heal themselves?

Depends: If it is just a simple inflammation or irritation from biting something hard and the tooth and patient are otherwise healthy i would say yes most of the time, but if there are other issues such as periodontal disease then it may lead to more probs.

They can: Think of the ligament as a cushion. It will recover from some, minor trauma. A partial avulsion of a tooth or trauma can heal itself once the trauma is removed. The ligament will respond well. The cells in the pdl also play a significant role in regeneration and tooth movement.