People say that.
We might get taught in schools not to say it. But to some people it's the natural way to talk. It's not ungrammatical to them.
It would probably be ungrammatical on a test of so-called standard English. But who decides what is standard? The test makers and those with socioeconomic power.
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
Yes, it is a valid English sentence, though the suggested versions are more natural ways of expressing the idea that zoos don't normally keep domestic animals such as cows.
We would use a sentence like that to say that there is a lack of something that we need and should have supplies of.
@FumbleFingers gives another example of a context where this construction is used.