Of course they're possible.
As to whether you'd want to use them, that's another question entirely. Each new auxiliary verb you use further narrows the temporal interpretation of the verb. After a certain point, it just doesn't matter 99% of the time. In other cases, it just sounds weird.
For the perfect continuous, that's likely because of the double be (has been being) that, like had had in pluperfect, even though it may be perfectly justified, often sounds better simplifying it (has been). Depending on the verb you need, you can partially avoid this slight cacophony by using a different auxiliar for the passive, like get, although that doesn't solve the convulatedness.
Continuous with modals will and would work exactly the same:
| It will have been getting written for hours |
| pres. bare inf. past.part. pres.part. past.part. |
| mod.fut. aux.perf. aux.cont. aux.pass. main |
Here I have a sentence, with the morphological form written on the second line, and the purpose of the verb (to generate the future, auxiliary to the passive, etc)
Modal will (or alternatively would) accepts any non-defective verb. Have fits the bill, and merely requires a past participle. That's been, which is used to form the continuous and simply requires a present participle. That's getting, which is one of the auxiliaries that can form the passive (you could also use being here). It needs the past participle of transitive verb, and written is just that.
Of course, if you use another structure other than modal will for the future, like going to it's even crazier:
| It is going to have been getting written for hours |
| pres. pres.part. bare inf. past.part. pres.part. past.part. |
| aux.cont. aux.fut. aux.perf. aux.cont. aux.pass. main |
I suppose if it's really important to emphasize that the fact that it is currently preparing to be in the process of being marked upon at some point in time prior to some time posterior to now, that works great, but really, it's overkill :-)
The verb tense depends on the time clause, the subject, and the object. Even though past perfect might be the most common choice for "By the + point in time" sentences, there are certainly cases where past continuous can be used.
In the first sentence, the most correct option is the past continuous, "were experiencing."
By the middle of 2008 many parts of Europe were experiencing the worst
economic recession for decades.
The object of the verb, economic recession, is something that takes place over a relatively long period of time (vs. a crash, for example)--it's an ongoing process. Also, the subject is vague and involves many separate entities, so even if a recession was definitely over for one (past perfect), for another country the recession could still be happening.
Past simple would never be the correct choice because the time clause "By the + point in time," indicates some separation between the point in time where it is located and the period of time when the event in the main clause occurred. You're saying that by this time, some event already happened and is over, or started to happen and is still happening.
In the second sentence, the correct tense is past perfect:
By the third month of the war rebel forces had taken most of the
province.
You wouldn't use past continuous here for contextual reasons. The sentence "By the third month of the war rebel forces were taking most of the province" doesn't make sense. "Most of" specifies an amount of the province under rebel control, implying that the outcome has already been seen.
Best Answer
Native speakers of English, or really any language, know how to use their verbs better than anybody else (and if they make 'mistakes', they make the right mistakes).
But you are really asking about conscious knowledge of their own language and the educated vocabulary used to describe it.
I can't say for other cultures, but in the US schooling system, there is some explanation of grammar with those terms you used in late elementary/early secondary school. But it is not emphasized or repeated later. As mentioned in a comment, most Americans learn more about grammar (and English grammar) when learning grammar of a foreign language.
As to you particular mention about two tenses, that's actually a bit of a terminological controversy, where most people informally think of English as having three, but they just would understand the formal vocabulary for tense as meaning English only has two inflected tenses (future and aspect are all periphrastic/modal/etc).
I distinctly remember in middle school (early secondary) some older student mentioning that they just learned about the future perfect in English, eg "I will have left for the store by the time you return", and thinking, wow, that must be some university level stuff. But it was never eventually mentioned to me in any English class (almost entirely devoted to literature and writing). And also eventually, I just learned how to do it naturally despite its rarity.
There are so many things in your native language that as a native you are just not aware of, but for non-native learners, an explicit rule makes things so much easier. Which preposition goes with a verb, the order of adjective roles, which article goes with a noun, the native speaker has no idea how to describe these, they just do it correctly without a thought.
To your title question, sure, most native speakers know the tenses past and present, they've heard things like 'present perfect continuous' but probably don't know exactly what it means, searching hard and failing from their memory of early grade school English.
Of course, ELU denizens, a very particular bunch, might track you down and forcibly explain every nuance of tense and aspect to you.