Here are the steps I would recommend:
1) Tighten up the privacy settings on all social media, block your dad and crazypants stepmom
2) DH needs to contact his boss, HR department, and possibly customer service to let the appropriate people know that his FIL is unstable and believes that you and DH are interfering with his computer/social media. Please add that he has contacted DH through work before and because he did not get "satisfaction" from his most recent interaction, he may try to either harass DH through work or contact the company to complain about DH as an employee. Note the time, date and escalation of Dad's hostile interaction with the customer service rep to highlight his instability.
3) Complete radio silence. You offered to send him his momentos. He (rude word)-ed all over your efforts. Do with those items what you wish and drop the subject. Do not contact him. Do not response to any calls, messages, emails, letters, telegrams, carrier pigeons, etc., that he may send you.
4) Prepare for escalation. Keep a log of all contacts from dad or crazypants stepmom. If your kids are involved in any sort of school program, extracurricular activity, etc., please let the program supervisors know that dad and crazypants stepmom are not allowed to pick up, contact or otherwise interact with your kids.
5) Don't worry about your supposed culpability in their computer issues. I sincerely doubt that your dad actually hired an investigator. And if they did, no authority is going to believe their story based on, "Well, the activity only stopped when they moved!" They would be laughed out of any sort of office where they went to file a complaint.
6) Snuggle your DH and your kids. You are a lovely person. You have a great family. Your dad is a bitter, deluded person who only knows to lash out through rage and crazy. Don't give him the satisfaction of devoting more energy or thought to him than he deserves, which is none.