These are called subject questions (as your first example) and object questions (as your second). Here we are talking about "do", so this is present simple and past simple tense. Let us take one example:
Paul wants to speak to him.
1) If we want to know who "him" is, "him" is the object and the question we will ask is an object question. There will be an auxiliary verb, which will come between the question word and the subject:
Who does Paul want to speak to?
2) If we ask a question about "Paul" (the subject), however, it will be a subject question, and then all we need to do is substitute the subject with a question word:
Who wants to speak to him?
Further reading with examples: in this grammar book
3) It is possible to add an auxiliary verb in subject questions, that has already been discussed here. To sum it up, it adds emphasis, as in:
A: Who wants to speak to him? Jack?
B: No.
A: Jim?
B: No.
A: Well, then who **does** want to speak to him?
More examples of that can be found in the link.
TL;DR It's fine to use more than one in a sentence.
If you're looking to explain it, note that you have two different subjects for the two verbs: "the reason is" and "it is". That's a good hint that you have two separate verb phrases. The two verbs aren't really constraining each other in any way — they could even be in different tenses.
In fact, you could use as many different auxiliaries in one sentence as you like!
I haven't yet told you that the reason why I've studied English all these years has never been because I've been enjoying it, but because my wife has never been committed to learning French.
Clarification
Note that neither of those instances of "to be" is traditionally considered an auxiliary (but see the appendix below for a different opinion).
The usual auxiliary verbs in English are indeed to be and to have. However, those can both also serve as main verbs: they can carry the main meaning of a verb phrase, not tense/aspect information. That's what they're doing here.
How do you know when they're auxiliaries? Because they will be followed by another verb, to make a compound tense. For example, has followed by a past participle makes the present perfect:
He has called his parents once a week since he moved out.
And is followed by a past participle makes the present passive:
Our little church bell is tolled to mark every wedding in town.
You can see that in your sentence, the only other verbs are "study" and "binds". Neither of them is by "is", so they're not relevant for any compound tense. So you have to be as the main verb.
Appendix
As others have noted, some grammar systems call "be" an auxiliary no matter what. The one @P.E.Dant linked to below seems to be an example. But however you slice the cake, at some point you have to draw a distinction between "to be" as a support for another verb vs. as an independent item ("non-core auxiliary", for example?).
It's worth nothing that in any case, "to be" behaves oddly for a verb in English, and so do similar words in many languages. Here it's a copula: it doesn't express an action like verbs are supposed to do, but declares an identity. You will see different grammar systems analyze it in very different ways.
The good news is that no matter which analysis is correct, you're okay to use more than one in the same sentence!
Best Answer
To start with, you need to understand about performative utterances — sentences which don't just describe something, but actually do something. For example, if I say "I apologize", then I am not merely saying that I apologize, but rather, I am apologizing simply by saying that I'm doing so.
Declare can be used for ordinary statements ("He declared that the meeting was a waste of time"), but it can also be used for performative utterances. "He declares the meeting open" is technically ambiguous, but it almost certainly denotes a performative utterance; it means roughly, "He opens the meeting [by announcing that it is now open]."
OK, with that background, we can start looking at the grammar . . .
In this version, declare is taking two separate complements: a direct object ("the meeting") and an adjective ("open") whose subject is that direct object. If it helps, here are some more sentences where the verb has a direct object plus a predicative complement:
(Note that these predicative can be either adjective phrases or noun phrases.)
As I mentioned above, this version is ambiguous between a performative reading ("He opens the meeting") and an ordinary statement ("He states that the meeting is open"), but it most likely means the former.
This version is pretty similar to the first. As before, declare is taking two complements — a direct object ("the meeting") and, this time, an infinitival clause ("to be open"). If it helps, here are some more sentences with this sort of structure:
Grammar-wise, I think that this version sounds much more like it's talking about an ordinary statement than about a performative utterance; but the concept of opening a meeting via performative utterance makes so much more sense than the other, that I can't really decide how it should be interpreted.
(I'd welcome other people's thoughts on this one.)
In this version, declare is taking a single complement: a declarative content clause ("[that] the meeting is open"), indicating what "he" is declaring. You don't seem to be asking about this one, so as far as the grammar is concerned, I'll leave it at that.
As far as the meaning is concerned, however, I think that this sentence really only works if we're talking about an ordinary statement, as opposed to a performative utterance. For that reason, I don't think it's at all synonymous with "He declares the meeting open."