You don't use inversion with imperatives, because inversion involves changing the standard order of subject/verb to verb/subject. Imperatives don't have an explicit subject (although "you" is implied) so there's nothing to invert. So, example "b" is correct: "Neither go there nor call her."
In your second question, yes, you invert after "not only", so "a" is correct (Not only are they noisy, but they're rude.). The non-inverted example "b" sounds very strange to me.
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
All three sentences are correct, but they can carry a surprisingly complex subtlety of meaning.
"I'm going for [something]", in this context, means I am departing to do or obtain that something. In other contexts, it could instead mean I am attempting to achieve something ("I'm going for the record"), or I am choosing an option (given a choice of desserts, "I'm going for the ice-cream"), but neither of these alternative possibilities applies to "going for a swim".
"I'm going to (have|take) a swim", on the other hand, mean that you are planning to swim. There is no implication that you'll have to go elsewhere in order to do so, as there is with "I'm going for a swim". In the specific case of swimming, this distinction is pretty meaningless, but if you replace the activity with, say, napping, it gains some relevance: "I'm going for a nap" is appropriate only if you are napping elsewhere, while "I'm going to (have|take) a nap" would also be appropriate if you're going to nap right where you are.