I think these two sentences are correct and make sense.
While I am wondering if the following two sentences are correct after omitting the subjects:
Any more references or examples about such situation?
The short answer to your question is no, it's not correct; you need to include the subject or pronoun again.
Even though English is considered a non pro-drop language, pronouns and subjects may be omitted in certain cases. So as far as I see, whether you are or are not allowed to omit subjects and pronouns in a language is a matter of degree in most cases, not an absolute dichotomy of yes and no.
It's grammatical to not repeat the subject or include a pronoun in certain cases. Some examples are after conjunctions like "and", "or", "but", yet, nor:
He went to college and (he) got a degree.
He went to college but (he) dropped out.
He promised to come, yet (he) never did.
In some cases, such as for "yet" it's much more idiomatic to repeat the subject or pronoun:
The fuel was running low, yet (it) was enough to get us to our destination.
In some cases like "nor" it may sound strange to many people to not to repeat the subject:
I will not cheat nor lie.
(some would insist on ", nor will I lie", it may sound wrong or archaic to some people)
In cases of some conjunctions that work in multiple groups of words, what are called correlative conjunctions, the subject or pronoun does not have to be repeated in many cases. These are conjunctions like:
either ... or ...
neither ... nor ...
not only ... but ...
Note that the second part of these conjunctions, "or", "nor" and "but" are the same ones listed further above. However some sources such as this one give others where it appears that the repetition of the subject/pronoun is necessary.
In the cases of "if" and "when", which are also conjunctions, it's just simply the case that you need to repeat subject/pronoun. You can try to pinpoint a rule that explains why this is, such as that coordinating conjunctions are OK but subordinating conjunctions are not, but I wouldn't trust this rule. "So" is recognized as a coordinating conjunction and as far as I can see you can't omit the subject/pronoun after it.
Interestingly, subjects and pronouns are dropped in informal English all the time, such as when saying:
(I am) Going to the gym. (I will) Be back in an hour.
Which is totally fine informally, but this isn't really considered formally grammatical.
✘ 1. Cross the task if has been done.
This sentence is ungrammatical as it stands. There are two way it can be corrected. The first is by adding a noun or pronoun, but the second is by actually removing even more components:
✔ Cross the task if it has been done.
✔ Cross the task done.
The second version does not mean the same thing as the first version, but it is still grammatical—albeit unusual. It would generally be assumed to mean cross the task that was done.
(Note that I'm not sure what the meaning of any of these sentence is. I don't know what crossing a task would involve. However, I'm only concerning myself with the syntax here, not the precise meaning of the sentence.)
✘ 2. I often swim when go to a gym.
This sentence is also ungrammatical. Like the first, it can also be corrected by inserting a noun or pronoun. Unlike the first, a simple change in verb form will also work:
✔ I often swim when I go to a gym.
✔ I often swim when going to a gym.
As with the first sentence, the second version allows it to be grammatical without using a noun or pronoun.
So, doing nothing other than removing the noun or pronoun will make each sentence ungrammatical. But there are ways of changing the sentences so that the removal of the noun or pronoun is okay.
