Can inserting and removing my Invisalign retainers cause trauma to teeth with a slight mobility? When is a permanent retainer recommended?

Wear it. It takes months for stability to be achieved. Because clear aligners are not quite as good as braces in achieving optimum alignment, retention is always a concern. Teeth should NOT feel mobile a week or two out of active treatment. If they do, return to your treating Dentist or Orthodontic Specialist for review. In the meantime, try to wear retainers as close to 24/7 as possible.
Maybe. If you have had your final alignerfor a while and all the teeth have not settled in, some form of fixed retention might be indicated. Ask your orthodontist what (s)he thinks.