In the following sentences, do we need " , and " to clarify that both verbs are for one subject?
1- " Alice went to the restaurant, and ate lunch".
2- "Alice went to the restaurant, ate lunch, and got back home"
Learn English – Using one subject for two verbs
grammaticalitysubjectsverbs
Related Solutions
I don't know a name for this. There might be one, but there isn't a one-word name for everything you could possibly do. I'd call it "one subject with multiple verbs".
As to whether it's legal: yes, certainly. It's very common. It's an effective way to express a connection between past and future events. "The committee started its work yesterday and will finish tomorrow."
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
In English, two verbs that refer to the same grammatical subject are normally connected with "and", and the two example sentences in your question are correct.
(I consider these sentences as examples of American English. In British English, there is usually no comma before "and", but see also Commas, subject and verb on English Language & Usage SE.)
There are languages that use "and" less frequently (e.g. 和 in Standard Chinese), but if you left out "and" in the first sentence, a native speaker of English would normally feel that the sentence is incomplete until you add "and [x]".