- What's the main thing a paramedic does have to do?
- What's the main thing a paramedic has to do?
How can we use the auxiliary verbs after such sentences like that?
1) What's the main thing a paramedic has to do?
Answer: A paramedic has to respond to emergency situations. Comment: that's a normal use of the verb have to do in the third person in the idiomatic usage of: to have to do something.
2) What's the main thing a paramedic does have to do? [implication: If he doesn't have to do A, B or C, for example.
If you are having a conversation with someone, and you claim a paramedic has to do certain things, and your interlocutor says you are mistaken about what a paramedic does, you might then use the emphatic auxiliary as in 2).
Another example of an emphatic auxiliary: Person One: He cleans the whole house everyday, he claims. But I don't think he does do that. [notice, an additional emphatic verb]
Person Two: Yes, well, he does claim that every time I go over there and see how dirty it is. But claiming something does not proof it as a fact.
In writing: Historians do proclaim certain truths when, in fact, there may be no concrete proof for their interpretations.
Emphatic uses of auxiliaries are found in writing and speech.
1: What's the main thing a paramedic does have to do?
2: What's the main thing a paramedic has to do?
Example #1 above is only valid in contexts where the speaker is responding to having been told that a paramedic doesn't have to (isn't required to) do certain things - in which case does must always carry very heavy stress.
Also note that the auxiliary have would almost always carry at least secondary stress, and be pronounced haff with a "hard" final consonant rather than the soft one in have. This is a special use of the verb form to have to [verb] (always followed by an infinitive verb), where it means be obliged to. In Past Tense contexts it's usually pronounced hat rather than had, so arguably it's actually a "different" verb, rather than a specialized sense of the existing verb.
It would require quite an unusual context for #2 not to require at least some level of stress on have (the stress indicating "obligation"). For example, if the conversation was focused on the fact that paramedics have a lot of "idle time" when they're just sitting around waiting to be called out to an emergency, in which case the activities "available" to them (to pass the time) might include things like playing cards together or browsing the Internet on their smartphones.
Another way to illustrate the significant syntactic as well as semantic difference between the "obligatory" sense and the more general case is to consider...
3: I don't have anything to do
4: I don't have to do anything (usually pronounced haff)
...where #3 means there are no (appealing) choices of action available to me (effectively, I'm bored), whereas #4 means there is nothing which I'm obliged to do.
In the above examples, we can tell which sense is intended by the word order, but given a written utterance such as...
5: These are the toys I have to play with
...with no indication of stress or the pronunciation of the final consonant in have, it's ambiguous whether the speaker means...
5a: I must play with these toys
5b: I have these toys available to me [if I wish to play with them]
First of all, one of your sentences is unsuitable for this context, as it contains the verb "to do" twice:
What's the main thing a paramedic does have to do?
You would only use the above to counter a previous statement made about what a paramedic doesn't do. If you mean it to be interchangeable with your other example then it is wrong.
It should be:
What's the main thing a paramedic has to do?
"Has" is the verb, and "do" is the auxiliary verb in this example. You don't need both, you could simply say:
What's the main thing a paramedic does?
The difference between asking what someone does and what they have to do is that technically "has to" implies some degree of obligation.
For example, if a child said:
I clean my room.
You would understand this is something they do, perhaps voluntarily, although that is not specified. On the other hand, if a child said:
I have to clean my room.
This carries the implication that they have either been made to clean their room by their parents, or possibly that they are compelled to clean it, maybe because it is in a terrible mess.
Having said that, in certain situations the difference is irrelevant, and I feel your example is one of those situations. What we do in our jobs is both something we choose to do and something we must do if we are under a contract of employment. Therefore I would say both "what's the main thing a paramedic does" and "what's the main thing a paramedic has to do" mean the same thing, idiomatically.