May I ask "What haven't mammals got?" assuming the "cold blood" as an answer? Or, similarly "What don’t most mammals eat?"
Yes, special questions, or wh-questions as I know them, can take negative forms. The Cambridge link notes
When we ask negative wh-questions, we use the auxiliary verb do when there is no other auxiliary or modal verb, even when the wh-word is the subject of the clause
In your example, have serves as an auxiliary of got. [A whole treatise could be written on got and gotten and their differences (also see at EL&U Should I use “got” or “gotten”). I think most style guides will broadly agree that in both British and American English, got is implicitly less formal, but it isn't wrong per se.]
Your question could be expressed as any of the following:
What haven't mammals got?
What don't mammals have?
What do mammals not have?
What is it [that] mammals don't have?
alt. What do mammals lack?
That said, regardless of how this question is phrased, it would be exceptionally broad. There are innumerable things mammals don't have— scales, feathers, magical horns made of rainbows, someone worth voting for in the upcoming U.S. presidential election. As such, you would want to rephrase the question to narrow the focus.
What are some types of thermoregulation not characteristic of mammals?
What are some things mammals don't eat?
Yes, this is possible, see for example, the title of this Ne-Yo song:
or, one of my favourite memes:
the formula is from quantum mechanics ...