SQL Server 2016 full backup and log shipping and newbie questions
once the restore on the secondary has been done, can the trn files be
  deleted?


Yes you can but you want to consider keeping files for certain period of time depending on your recovery point objective.


  if I wanted to combine the transaction log shipping with a
  full/differential backup plan


Full and differential backup will not break your log chain.  

Make sure you do not take any log backup outside log shipping. All log backups must be done through the log shipping. Because that will break your log chain and hence log shipping will fail.