You aren't talking about the parties together as a unified group, but how they feel individually. You generally are always doing this when you use neither.
This may help you understand:
Neither [one] of the parties is concerned with Egypt
Usually the one is understood and left out of the construct.
I'm not sure if the below is correct, but here goes:
Let's say you had a total of four parties, and for some reason, they were afflilated in two groups, i.e. party A and B, and party C and D. Then you could say:
Neither two of the parties are concerned with Egypt.
You can see how this is confusing unless there's a strong context to help, so I would really avoid using neither in this situation, even if it is correct.
The last is very odd, without context. There are "and" phrases which we understand to mean a single item:
Fish and chips is my favourite meal.
If I changed this to "Fish is my favourite meal, and chips are my favourite meal", the meaning has changed. "Fish and chips" is a singular item.
Your example isn't like that. So (5) is at least very odd, and I'd say ungrammatical.
The other are ok, but 4 is odd, and could probably be misunderstood, at least on first hearing.
As a rule of thumb, if you can split the sentence into two coordinate clauses then the subject is plural (Source). However, the situation you describe is awkward, as is the producer/director one in the comments. So avoid it if possible. It is nearly always possible to rephrase.
This is a confusing situation, so more writing to explain would help.
I would write:
In his role as a doctor and as a patient, he is a good man.
"Role" is a key word here, it emphasises one man with two positions.
Don't say "The director and producer of the movie was not present." Say "Speilberg was both producer and director, but he wasn't present." It is hard to think of a situation in which you would have to use a plural subject as singular.
I don't recognise a rule based around articles. The "rule" is "verb agrees with subject" and 1-4 all obey this rule.
Best Answer
The contraction What's in your example expands to What is.
CGEL (pages 505-6:) tells us in Section 19 under Interrogatives:
Later in the same section, it also tells us:
If there is an expectation or prior knowledge that the pronoun refers to a plural subject, the default singular value may not be expected. In your question (as edited) there are multiple items in the bags. It may therefore be proper to use the plural form of the verb in:
However, there is an important distinction between proper or grammatically justifiable and idiomatic. In common speech and usage, the question would very seldom be posed in that way by a native English speaker. In particular, the contraction What's is used in numerous contexts in which a plural verb might be "correct," but in which the construction What are would elicit puzzlement from the listener. For example:
When what is used as an interrogative pronoun and subject, it usually takes the singular form of a verb. To take the plural form, your example question could be reformulated so that what functions instead as an interrogative adjective:
(A detailed answer to this and related questions can be found under the question Which is the correct question (“Who has” vs “Who have”)?)