This is atrocious English
But you're correct, it has become common in colloquial speech, especially among some subcultures. The tendency to drop the verb "to be" and its counterpart behavior, the tendency to never conjugate the verb, stem from U.S. subcultrues in the 1970s, which were derived from the "Jive" lingo from the African-American jazz culture of the 1930s and 1940s. Both behaviors are ungrammatical (unless the world today has decided to become accommodating, language is fluid).
An example of dropping the verb comes from the 70s popular TV situation comedy, "Diff'rent Strokes."
What you talkin' 'bout Willis?
An example of not conjugating the verb comes from a more recent experience. While helping a friend prepare to test for his commercial driving license, he would chant through the pre-drive test and check procedures like this:
We be looking at the tire guage depth...
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, I would change both sentences from "that" to "it." It's up to you has two definitions: the first that something is someone's decision, and the second that something is someone's responsibility.
If up to somebody is followed by an infinitive, it indicates responsibility.
That's up to you is usually used for decision, not responsibility. So in sentences with an infinitive structure after up to you (like both of your examples), I would expect it and not that.