They are not exactly interchangeable, though some of your example pairs are more similar than others. Though as a native speaker I often hear and see present simple and present perfect used interchangeably in different situations, but I think that that is often sloppy. Their meanings frequently overlap, but they convey different information.
The key difference is that present perfect clearly describes something that happened in the past but continues to be relevant today, while present simple describes something that is true right now, with no reference to the past at all. These produce somewhat different meanings:
A1: There has been a rumor that the president is ill
A2: There is a rumor that the president is ill
A1 means that the rumors existed in the past, and implies (but does not require) that those rumors are still around (if the rumors are definitely no longer around, were would be preferred). A2 means that, right now, that rumor exists and does not reference the past at all. They express different things, even if they both have the same implication for the status of rumors about the president's health in the present.
B1: There has been a mistake
B2: There is a mistake
These are harder for me to parse because "mistake" can be used in a couple of different ways. I might use "there is a mistake" to describe a misspelled word in something I've written-- I'm describing the misspelled word as a mistake, which currently exists. But I could also describe the action that caused the misspelling as a mistake-- "there was a mistake" might use "mistake" to describe an error in my typing or in how I thought the word was spelled.
My immediate reading of B1 and B2 is that B1 is probably describing an action (likely completed in the past) that caused some consequence which is still relevant, while B2 is probably describing a thing that is currently in error rather than the action that produced that thing (the misspelled word is itself a mistake, which is different from me saying that I made a mistake and now the word is misspelled on the page in front of us). These would not be interchangeable, though I'm sure that there are examples of similar sentences that would be more able to substitute for one another.
C1: He has been taken to hospital
C2: He is taken to hospital
C2 is not correct. If he is currently in hospital, then the "taking" has been completed and so should be conjugated in the preterite tense: "He was taken to hospital." C1 strongly suggests that he is currently in hospital because the present perfect indicates that his having been taken there is currently relevant, which it would not be if he had already left. "He was taken" does not have this suggestion, because it makes no claim to being currently relevant information.
D1: I have been on holiday for two weeks
D2: I am on holiday
These both mean that you are currently on holiday, and if that's what you want to express then both sentences will work. But if the fact that you have been on holiday matters ("I didn't see that email, I've been on holiday for two weeks") then D2 would not be sufficient.