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

Solution 1:

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.

Solution 2:

'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'.

.......

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

........................

'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.

Solution 3:

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