"Don't mind what she does tomorrow" but not "Don't know what she does tomorrow" Why?

'I don't mind what ...' can have two meanings, and these are informed by the constructions used.

'I don't mind what you are doing' usually means 'I am not troubled by the things I know you are doing'.

'I don't mind what you do tomorrow' means 'Whatever you choose to do tomorrow is OK by me'.

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'I don't mind what you will do tomorrow' is unidiomatic and thus has no default sense.

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'I don't know what she does' is idiomatic for 'I don't know what her usual practice in these situations is / I don't know how she earns a living'.

'I don't know what she does tomorrow' must have the sense 'I don't know what the timetabled practice for tomorrow is'.

'I don't know what she will do tomorrow' is transparent.

The question posed is about 'the acceptability of ‘will’ in different environments'.

In one of these environments I believe that language comes up against what is currently termed ‘cognitive dissonance’ - a conflict not of logic, as such, but a conflict between personal value and personal action. It is the conflict of being obliged to do something one does not, personally, feel one should be doing. (Like uttering something that it is just not possible to utter.)

This arises, in this case, when someone is obliged to picture themselves in the far future, looking back towards the near future, in order to report a future happening as though it was a completed event.

I believe this is why, in English, we have a situation which some would describe as “possessing no future tense”. The reason we do not have what they term “a future tense” is because no such thing can exist. And many languages, I am informed, are thusly. My suspicion is, they all are.

In order to confidently make predictions about the future one has to project oneself into the far future, then look back at the near future (as though it were a past event) and report the occurrence of that event as though it had been completed, had been documented, and could be looked upon as past history.

Some have not understood what is involved in making a future prediction and their misunderstanding created the controversy of the Waw Conversive, which is ably dealt with by Robert Young in the preface to his Literal Bible. He blasts it to smithereens.

‘I don’t know what she does tomorrow’ answers the question ‘What does she do tomorrow ?’

‘Does’ is a continuous matter. ‘Does she like ice cream ?’ ‘Does she visit the theatre ?’ ‘Does she have brown eyes ?’ It reports either a continuous state or a repeated activity.

The possible answers to ‘What does she do tomorrow ?’ are many and various :

‘I haven’t a clue; let me hop in my time machine and I’ll be back five minutes ago and one of us will tell you’ is one of the least exasperated outbursts which could result.

I would have to predict, in the future, what her state or her repeated actions will be and then I have to report them with the same degree of certainty that I report what I had for breakfast three hours in the past. Her eyes could be bloodshot, no longer brown. She may now hate both ice cream and theatre.

So my answer, properly punctuated, is ‘I don’t know what she ‘does’ tomorrow,’ (!)

How can I know ? How can I answer your question ? So ‘does’ is in quotes, because you said it, not me. And ‘know’ is in italics because I really do want to get across to you that I don’t know !

I don’t mind what she does tomorrow - my mind is completely detached from the future event and I am comfortable with that. No problem. She can do what she likes; polish the time machine, maybe. (Spoiler alert : we need it again.)

I don’t know what she does tomorrow - haven’t a clue, so I need not engage my sensitive cognitive faculties and they, thus, will not become dissonant. I am totally disengaged from tomorrow. It can be worded 'I don't know what she will do tomorrow' and it doesn't matter. I still do not have to predict anything.

I don’t mind what she will do tomorrow. ‘Will’ is her will, not mine. She will do fifty somersaults tomorrow (but only if she wills to do it). If somebody else wills to shoot her between the eyes, she won’t be doing a thing. So the blue button gets you into the future and the green one gets you back again (don’t touch the red one). Once you’re back and can tell me what she (and everybody else in the world) will be willing to do tomorrow - at that point I’ll tell you whether I mind or not.

We are creatures of the present (with memories). This is all we know. It is all we can know. It is all that language will let us report. And if we try to bend language, we become dissonant

The following sentence is unidiomatic in English:

I don't mind what she will do tomorrow. NO

But the sentence can have a future meaning if we add the auxiliary "will" before the verb mind

I won't mind what she does tomorrow.

Today, the more common response would be

I don't mind what she's doing tomorrow.

Cambridge Grammar contains a note on the usage of mind

Warning:
When we refer to the future, we use present (not future) verb forms after mind:

I don’t mind what day they come and stay as long as it’s not Tuesday 12th because I’m away.

Not: … what day they will come and stay …

I suspect that do not mind, is a relatively recent development stemming from its more polite and formal equivalent would not mind, which is used to make polite (and sometimes hesitant) requests

A: Would you mind if I open the window?
B: No, but I should mind if you were to leave it open all night.

From Fairy Birds from Fancy Islet, or the Children in the Forest, dated 1846, the phrase I should mind has a future meaning. Today, the “I don't think” would probably be omitted, and “I should mind in the least” would be replaced with I don't mind in the slightest

“I do not think,” said Alice “I should mind in the least having my wings nailed on.” “Should you not?” said Jim, (half ashamed of his cowardice;) “ yes, but you would, though.” “No.” said Alice. “I am sure I shall not: I will have the wings nailed on directly, if the grey bird think it the best way.” […]

“Why,” said Jim, “you know very well, that your saying you would not mind having the wings nailed on your shoulders, must make me appear very cowardly, so if you did not feel afraid, you should not have said so, for my sake.”

For the "don't know" construction, the present progressive tense, which is also used to express future meaning, sounds far more natural and logical than “I don't mind/know what she does tomorrow”

I don't know what she's doing tomorrow
I don't mind what she's doing tomorrow

The major difference between the predicates distinguished here with Wh-clause complements is that

• not know belongs to a class of predicates that takes a Disjunctive Wh-clause complement

whereas

• not care (in the US) belongs to a class that takes a Conjunctive Wh-clause complement.

The difference is in the pragmatic nature of the Wh-clause; conjunctive clauses are factive --
they presuppose the identity and truth of their complement. E.g,

• I am aware of/They'll be surprised by/He doesn't care what she does tomorrow.
  In this example, the clause refers to the set of all actions that she is to do tomorrow,
  as a settled matter; it's called conjunctive because the set consists of A and B and C and ...

while disjunctive clauses are indefinite, and presuppose nothing, E.g,

• It's a mystery/I wonder/They don't know what she'll do tomorrow.
  In this example, the clause refers to an unknown set of possible events or actions;
  it's called disjunctive because the set consists of A or B or C or ...

Since the conjunctive clause is presupposed, there's no need for a predictive modal like will. But the disjunctive clause is indefinite, and its truth set may be empty ("..., if anything"), so the predictive will is necessary.