Work out from the middle: you know that there are $3$ people who can sing, dance, and act. There are $10$ who can sing and act, but $3$ of them are already accounted for, so the shaded region at the top where you have $10$ should really have $7$: $7$ of the people who can sing and act cannot dance, and the other $3$ can. Similarly, there are $7$ who can act and dance but cannot sing, and there are $5$ who can dance and sing but cannot act.
Now consider the singers. There are $21$ of them altogether. $7$ of these can also act but cannot dance; $3$ can also act and dance; and $5$ can also dance but cannot act. These singers who can do something else as well account for $15$ of the $21$ singers, so there must be $6$ singers who can neither act nor dance. Similar reasoning, which I’ll leave you to try, shows that there are $6$ dancers who can neither sing nor act and $5$ actors who can neither dance nor sing.
If you now add up the figures in all seven regions within the three circles, you should get a total of $39$. Since there are $42$ people altogether, this means that there must be $42-39=3$ managers. Note that up to this point the directors and poets are red herrings: you can ignore them completely.
The last question is ambiguous. I can’t tell whether it wants the total number of people who can do exactly two things, the total number who can do at least two things, or for every pair of things the number who can do at least (or exactly) those two things. If it’s asking for the number of people who can do at least two things, we can start with the $3+7+7+5=22$ people in the various intersections of the circles. Now consider the poets: all of them are actors, so all of them can do at least two things. Two of them, however, are actor-dancers, so we’ve already counted them in the $22$. The other $3$, however, are actors who neither dance nor sing, so they weren’t counted and need to be added; this brings the total to $25$. Similarly, one of the $3$ directors has already been counted (as a dancer-singer), but the other two are dancers who neither sing nor act, so they have to be added to the total of those with multiple talents, bringing it to $27$.