This use of see is not idiomatic in Anglo-American English. The corresponding minatory phrase is
I'll get you!
This probably represents an underlying “I’ll get you back” = “I will repay you for this injury”.
That idiom suggests a line that Arnold Schwarzenegger's Terminator films have made into a paradigmatic threat:
I'll be back.
It may also be possible that “I’ll get you” represents “I’ll get you where I want you” = “I will cause you to occupy an unpleasant situation in which I would like to see you”. I suggest this because there is an old AAE idiom, little used today but once popular in melodramatic dialogue, which used see in this sense.
I'll see you damned / in hell / hanged / in jail!
This does not imply literal presence when the addressee is in the named predicament, but gleeful anticipation of that situation. There is a discussion of one common (and still current) variant at ELU.
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
too / either indicates that one person or things is (to be) treated the same as one that has already been mentioned. too is used in positive sentences:
either is used in negative sentences, but only if the first sentence is also negative:
If you have a positive sentence followed by a negative sentence, you use too, even though the second sentence is negative...
Adding forget to doesn't really affect the choice of adverb:
So, in your sentence, the two parts are:
too is the correct adverb to use, even though the second sentence is negative, because the first sentence is positive.
Note that, if both sentences are negative, you use either: