As it was said in the comments and in the first answer, the sentences without who/which in your examples are wrong.
But, speaking in general, there is really a rule that comes to mind when looking at your examples. This rule regards to the relative (a.k.a adjective) dependent clauses. This rule states:
You can omit the relative pronoun of a relative clause, ...
To me, it is very strange to see "both" used with 3 items. I am a native speaker and I can safely say that in every context I can imagine, it is only used for 2 items. Unless, it was posted by a foreign speaker or maybe it is acceptable for the programming/coding world. I would use "all three" as an alternative.
Only the Deepl translation is grammatical in English. It refers to "plain text and text inside tables and footnotes". I have no idea about the fidelity to the Russian.
The word "both" must be used with two items. The Deepl translation groups "text inside tables and footnotes" as a single item, and "both" refers to the ...
"Both" is used from two items, and it is confusing to use it with a list of three thing. I want to try an work out if two of the three things are really combined and count as one thing.
You can use "all three" for emphasis, or just omit it:
... for (all three of) body text, text in tables, and footnotes...
using "as well as" ...
We looked at the job applications in batch
This sounds like jargon. It might be understandable to the people involved but not to the general public.
We looked at the job applications in batches.
Is there a rule I can follow to decide whether I should put a determiner before the countable noun "batch"?
The rule is that "batch" should have a determiner, such as "the", "a", "three", "this", etc.
An exception might be if "batch" is used as an adjective. Then, "update the users ...
(1) is definitely not a usual construction. For one thing, it's is short for it is or it has, so we never say it's has.
If you have had your first family vacation in four years, use sentence (2) (but with It has).
If the family has not had a vacation in the last four years, say It's four years since our family has had a vacation.