Coordinative ellipsis with not and unparellel structure

I worked and did not play.

I worked and not played. ?

For some reason, when the ellipsis is used after the simple past form of the verb with no "to be" used before it, it just sounds weird to me. Is there something wrong with it, or is it just that it is not idiomatical? Or am I the one who's wrong?

I had to control myself and not cry in front of her.

I always thought that second part has been used independently and not connected to the "had to" part, for I thought there should be additional "to" like this

I had to control myself and had to not cry.

Or at least this: I had to control myself and to not cry in front of her. (It doesn't sound very idiomatic... is it acceptable?)

Is the "to" deleted? If it is, why is that?

And also, when I was trying to find the answer for this question, it was clear that everybody was asserting only the parallel form should be allowed for the ellipsis after the coordinating conjunction. For example:

Sarah is [a CEO] and [proud of her job]. - NP + AP

Is Jim [conservative] and [a closet Republican]? - A + NP

He is a crazy man and may drink a lot. -NP+V

These (only the three sentences directly above it) are from Wikipedia, and not regarded as being incorrect. But looking up at the other websites, I find that they say that it is wrong, even though many people, including me, use it this way. They say that it should be parallel like this:

Sarah is [a CEO] and [a person proud of her job]. - NP + NP

Is it supposed to be always parallel?

For these to be correct you must be able to attach the subject to both of the phrases surrounding the coordinating conjunction individually, as you stated. There is no reason for a sentence to not be correct if it follows this rule.

• I [worked] and [did not play].
• I [worked] and [not played].

The first sentence is correct because you can split it into I worked and I did not play whereas splitting the second sentence gets you I not played which is ungrammatical, therefore the original sentence is ungrammatical.

• I had to [control myself] and [not cry in front of her].
• I had [to control myself] and [to not cry in front of her].

Both of these are technically correct, however the first is much more fluid. The second sentence is grammatical but can take on a slightly different meaning than the first sentence. Here, not crying in front of her is implicitly a separate event from controlling yourself because the to after and "resets" the function of the coordinating conjunction; the and implies a relationship between the two phrases, but inserting to breaks this.

• Sarah is [a CEO] and [proud of her job].
• Sarah is [a CEO] and [a person proud of her job].
• Sarah [is a CEO] and [is proud of her job].

These are all grammatical based on my first point. I have added a third version because this is one of the more common ways this sentence would be used. Many would say the first is too short, the second is too long, but the third is just right. Past the rule I stated, there isn't a definitive way to pick the truly correct version here. It's all user preference because they're all grammatical.

A little bit on parallel and unparallel structure:
Both structures are correct and can be used interchangeably. In my explanations above, I was stressing parallel structure since it is the better way to go. However, unparallel structure still works. Here is an example:

I bought a new refrigerator, dish washer, and the oven will be delivered tomorrow.

This is unparallel because the sentence can be split into two parts, I bought a new refrigerator and I bought a new dish washer, but the third part doesn't follow the I bought a new pattern. The third part grammatically fits but doesn't keep the pattern. This structure is used quite often, but it isn't the more correct way to write.