I'm struggling with "will I" in this sentence. Is it grammatically correct?
I know that "shall I" would be fine, but does "will I" work too?
In first person questions, use shall I for things you have control over, and will I for things you don't.
Shall I go to church?
Will I go to jail?
Unless there are some very unusual circumstances connected with opening the box, you should use shall I.
I perceive shall I? as less tentative than should I?.
If I say:
Shall I open the box?
I'm essentially suggesting that I open the box, and asking whether you agree.
If I say:
Should I open the box?
it's less of a suggestion, and more asking your opinion one way or the other.
I think the reason you are struggling is twofold.
The first problem is with the words shall and will. While in the past did have an annoyingly confusing conjugation, they are now mostly used interchangeably in English. See: Oxford Dictionaries' comment on this which says:
In practice, though, the two words are used more or less interchangeably, and this is now an acceptable part of standard British and American English; however, the word shall is now seldom used in any normal context in American English.
FWIW, the traditional rule is that the conjugation:
I/we shall, you/he/she/they will
Is used to form the future tense,
I will, you/he/she/they shall
Is used to indicate determination.
Again, this is annoyingly odd because will and shall flip meanings depending on whether they are first, second or third person.
To offer my personal experience FWIW, I think the word "shall" in its various guises often has an archaic, or perhaps pretentious ring to it, especially in the United States, and given how odd it behaves, I do not lament its demise. However, I think many will disagree with me of that point, though, certainly for American English, the quote from Oxford Dictionaries does support this contention.
Secondly, I think the sentences itself is rather confusing. What are you trying to say?
"Am I going to open the box? No you mustn't." might be one meaning you are seeking, and honestly, using the wording you do, if this is the intent, is rather clumsy.
You might also be going for "Should I open the box? No you mustn't." Which has a somewhat different meaning, and is not really what your original sentence means.
So, based on these two points, I suggest you reword your sentence for a little more clarity.
You can use shall, should, would, or will in your example sentence
_____ I open this box?
The choosing of the word depends on what you want to portray to your audience.
Shall is used to make/ask for a suggestion also could be used to ask questions politely
Shall I open this box?
Should is often used to give/ask for an opinion, to make a suggestion, express a preference or an idea.
Should I open this box?
Will is used to show/ask for desire, preference, choice or consent
Will I open this box?
Would is used to show/ask preference
Would I open this box?