something or anything

I found recently following sentences in a novel and I think according to
school grammar I'd learnd years ago that 'something' should be 'anything'
and thus the sentence can be incorrect.

"Bring your son by this afternoon. But please have a physician check him out first.
Just to make sure we're not missing something."
(Dr. Death by Jonathan Kellerman, Ballantine Books, 2000. Page.199)

I would be grateful, if you could answer my questions:
1. Is 'something' as well as 'anything' correct in this case?
2. If it's possible to use 'something',
what difference between 'something' and 'anything'
would be there on nuance or meaning?
3. If the speaker were mistaken by saying 'something', which effect you think
the author would have intended with this mistake?

1. Yes, both would be correct.

2. In this case, 'anything' would imply an unspecific thing over an very large range of possibilities. The speaker has nothing specific in mind that would be missed. 'Something' would imply the speaker has something specific in mind over a smaller range of possibilities. The speaker probably thinks or knows that something would be missed, which is why the speaker requests the son to go see a physician. More context would be needed but 'something' would be the 'correct' choice here.

3. Again, more context is needed but presumably the speaker, if a doctor, misdiagnosed or misinterpreted something leading the speaker to think that there's something specific that would be missed. Neither is 'wrong' really.

I appreciate your responding quickly and
explaining in detail and think I got the solutions.
Regards.

In the example you give there's virtually no difference in outcome.
Situations where 'something' is a better choice than 'anything' would include:

"I'm hungry, I want to eat something."
"I see something coming over the hill."

opposite:

"Don't get mad at me, I didn't do anything!"
"Fine let's just do that, anything would be better than waiting here even longer."

you also have to remember that it is dialogue and not every one who says something will say it correctly. especially in your native tongue.
in this case it seems like they are making a joke.
its as if the boy they are talking about may have some strange problem and they are hinting at it. but im not sure.