I think the call tasks.get_all should have you covered to extract all information you would need.
None
The request body should look something like this:

{
  "id": [],
  "scroll_id": "b77a32d585604b098f685b00f30ba2c2",
  "refresh_scroll": true,
  "size": 15,
  "order_by": [
    "-last_update"
  ],
  "type": [
    "__$not",
    "annotation_manual",
    "__$not",
    "annotation",
    "__$not",
    "dataset_import"
  ],
  "user": [],
  "system_tags": [
    "__$and",
    "__$not",
    "archived"
  ],
  "include_subprojects": true,
  "search_hidden": true,
  "only_fields": [
    "name",
    "status",
    "system_tags",
    "project",
    "company",
    "last_change",
    "started",
    "last_iteration",
    "tags",
    "user.name",
    "runtime.progress",
    "type",
    "project.name",
    "last_update",
    "parent.name",
    "parent.project.id",
    "parent.project.name"
  ]
}

Tip: Anything that you want to extract from the UI, you can use the API. To find the relevant call you can always open developer tools (F12) and see what returns