To formalize what StoneyB said, computerized grammar checkers are only meant as guides because they don't know context, they only know rules. You really need a human editor or a place like this to figure out if you are writing / speaking something correctly.
In this case, your sentence is perfectly legit. What you are doing is asking a rhetorical question, because you, the author, are not expecting an answer to your question. Instead you are using your question to make a point.
Actually, they just added a canonical post about this very subject!
So can be used to say that another subject is doing something "as well" or "in addition." - you can keep things in one sentence with so but not with as well or in addition (so that's why it happens in speech often):
Vinay plays cricket, and so does Ashok.
Vinay plays cricket. Ashok does as well.
But if there's a second action, so takes on the meaning of because of that:
He can speak French, and so can speak German.
He can speak French and because of that he can speak German.
Don't use so at all, simply say
He can speak French and German.
The problem with this:
I seldom went to Christmas parties, neither did he.
is that while "seldom went" has the meaning does not go that often, it's not a verb in negative form, so neither doesn't correctly link back to it.
You can either change "seldom went" to a negative verb, add "he" to the subject of the sentence, or make "neither did he" a full sentence with "either". ("Seldom" is one of those words you generally don't want to overuse by using it in two sentences in a row.)
I didn't tend to go to Christmas parties, neither did he.
He and I seldom went to Christmas parties.
I seldom went to Christmas parties. He didn't usually show up to them either.
Best Answer
These are called subject questions (as your first example) and object questions (as your second). Here we are talking about "do", so this is present simple and past simple tense. Let us take one example:
1) If we want to know who "him" is, "him" is the object and the question we will ask is an object question. There will be an auxiliary verb, which will come between the question word and the subject:
2) If we ask a question about "Paul" (the subject), however, it will be a subject question, and then all we need to do is substitute the subject with a question word:
Further reading with examples: in this grammar book
3) It is possible to add an auxiliary verb in subject questions, that has already been discussed here. To sum it up, it adds emphasis, as in:
More examples of that can be found in the link.