The ID of a record literally never changes.
One way to put that Lead's ID into the Opportunity or the Contact is to have a formula field on the Lead with the formula: "id" (no quotes). But it will be a text field , a formula cannot evaluate to an ID datatype. Then Lead Conversion can map that text field to a text field in the Contact and/or Opportunity. Since it is text, displaying the Lead from a Contact or Opportunity page, or pulling data from the Lead, will require a bit of URL hacking or VF or Apex
Also note that the typical User cannot see converted Leads (no Read access to the records). There is a permission that enables a user to view and edit Leads; it's both, or neither. A Flow running as System should be able to see the record and do the heavy lifting.
