Current implementation of wikilambdasearch_labels API allows for all of this. See the API updated documentation:
https://www.mediawiki.org/wiki/Extension:WikiLambda/API#list=wikilambdasearch_labels

Responding to each point:

only return objects with that exact type:

api.php?action=query&format=json&list=wikilambdasearch_labels&wikilambdasearch_language=en&wikilambdasearch_type=Z40

return objects of that type and functions that return objects of that type

api.php?action=query&format=json&list=wikilambdasearch_labels&wikilambdasearch_language=en&wikilambdasearch_type=Z40&wikilambdasearch_return_type=Z40&wikilambdasearch_strict_return_type=true

return objects of that type, functions that return objects of that type, and functions that return objects of any type

api.php?action=query&format=json&list=wikilambdasearch_labels&wikilambdasearch_language=en&wikilambdasearch_type=Z40&wikilambdasearch_return_type=Z40

The result should rank them together in a single list. Each result should state its type (so we can mark up functions)

The results are returned in a single list and each element contains information of its type and return type:

{
    "batchcomplete": "",
    "query": {
        "wikilambdasearch_labels": [
            {
                "page_namespace": 0,
                "page_title": "Z111",
                "page_type": "Z111",
                "return_type": null,
                "label": "foo",
                "is_primary": 1,
                "page_id": 0,
                "page_content_model": "zobject",
                "page_lang": "eu"
            }
        ]
    }
}

I believe this task should be marked as done.