Both are grammatically correct.
It's a subtle point, but referring to "the study of Something" makes it sound like you are referring to the entire field of research. Saying "By studying Something" makes it more clear that you mean the specific studies you have conducted.
You might make this even more clear by saying "Based on the studies Title X, Title Y, Title Z, we can conclude..."
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 word 'then' has many meanings. In some of its meanings, it is perfectly grammatical at the end of a sentence.
For instance, from definition 3.b from the Merriam-Webster online dictionary (here)
Another example is using this meaning (here):
The Macmillan dictionary describes the meaning of the OP's question directly:
This last one was the meaning of 'then' in the OP's post.
Person C means "As a result of what you just told me, I conclude that 'you both are really lucky'."