Should foreign words used in English be inflected for gender, number, and case according to the conventions of their source language?
Is there a general rule for whether, for, example, foreign nouns and adjectives used in English should be inflected for gender, number, and case as they would be if the entire text were written in the source language? If not, what form should the foreign words take? Might the rules be different for gender, number—i.e., plurals—and case?
Should the English surrounding the foreign word be made to correspond with the foreign gender, number, and case?
Examples:
The data tells us that the earth is getting warmer.
I hadn't realized that my bananas would be flambé.
I have no desire to read another poshloe book by a novyĭ russkiĭ.
I, too, said na'aseh v'nishma at Sinai.
I asked a khosheve rabbi this question.
Related:
Re: Names
Plurals of foreign words—only addresses the question of number and does not give a categorical answer
Preceding article in foreign words
Should I follow English conventions, or write what sounds better?
Use of plural pronoun to avoid mentioning of gender
Anglicized plural and zero plural
Which style of Latin plurals should I use?
Pluralisation of Latin Words
https://english.stackexchange.com/a/38113/98198
The answer is, unsatisfyingly, that it depends. Most native speakers aren't fluent in the borrowed language and so won't know the grammar principles there.
Sometimes things are borrowed exactly, like Latin sayings, and stand alone with no possibility of declining, like 'ceteris paribus'
Sometimes a simple thing, like a plural, if easy, is declined, like Greek the singular for criteria is usually said 'criterion'. But sometimes that strict adherence to a foreign grammar is lost and 'criteria' is used for singular, 'criterias' for plural.
Depending on how unnatural or infrequent the foreign word is, the foreign manner may be preserved or not: 'My fiancée' for a female to be married is correct English not because English has grammatical gender like French but English allows an alternate lexical item (ignoring for the moment that it is a difference in spelling only). But 'The banana was flambé' is grating in English, so 'flambéd' is preferred, despite it sounding very unFrench.
The principle is that someone who has heard the foreign word repeats it to an English speaker. However clear that foreign word is spoken (by native speaker or not), the English speaker will assimilate it as best they can, ignoring what just doesn't work in English, and preserving possibly what does work.
So, for an arbitrary foreign word slipped into English, what form should it take? You probably want to preserve what you can and drop what just sounds bad. And then English speakers will do with it what they may. Early on, 'data' was plural, but nowadays, since 'datum' is almost entirely unheard of, different groups continue to use 'data' as plural (rarer) and others use it as a mass noun (taking a singular verb).
So there's no hard and fast rule; try to conjugate as much as you can, but once it is firmly in English, then English rules will apply.
You must, as always, write for your audience.
If you are writing for a technical journal where your audience is multi-lingual, then you should strive to get it absolutely right. That goes without saying.
If you are writing for the general public, then you would probably need to base your choices on the common usage for each word and phrase.
The use of data in your first example is problematic. If you are writing for a technical journal, you should use "the data tell us", since this is a plural form and your audience will expect you to use it. However, not everyone (in the UK) will know that data is the plural form of datum and so, when writing for the general public, it is common to put "the data tells us", and treat data as a mass noun.
As a corollary, very few native English speakers will know that fora is the plural form of forum (unless they have studied classics), so forums has become the accepted form in everyday English. You should only use fora if you have a justifiable expectation that your audience will know what it means.
In everyday BrE, we would probably use flambéd as the adjectival form instead of flambé, i.e. "I hadn't realized that my bananas would be flambéd." However, you would expect a good restaurant to get it right on their menu:
Bananes flambées (flambéd bananas)
I can't comment on your other examples since I have no knowledge of Russian, Hebrew or Yiddish.
Wherever possible, always get advice from the publishers that you are writing for and follow their house style guide.
Other parts of the question have been discussed extensively but I haven't seen much about the grammatical case so let me contribute in this area.
On cases
There's a more general problem with this approach that even in languages with cases, the case to be used does not necessarily correspond. So if you loan a word from, say, Russian into Czech (and those are close!), you may still need to decline it in a case that would not be natural in the original language in the same context because in its original form there's just no way for it to work. "About", for example, is expressed in Russian using accusative case with the preposition про, in Czech it is o with a locative. What's worse, accusative pro exists in the latter but means something else (for), and accusative o is something yet else (sparsely used). So using the Russian accusative would either mean breaking the preposition–case correspondence or constructing a nonsensical sentence.
I'm not a native speaker of English, and neither a language expert, so don't take me literally on this, but as far as I know, it actually has cases, subjective and objective. If you apply the above rule here, it tells you that you should convert the nouns to one of these, which will most probably be identical to each other anyway (and in many cases you can't use objective / oblique case anyway simply because there is none in the original language).
I can offer a direct example hinting against forcing foreign declension in English, or generally, forcing the use of a case which does not exist in the target language. My language is one of the few ones retaining a vocative. While there is a discussion whether foreign names should ever be declined at all (which usually ends in a case-to-case resolution) and some could not possibly be given a vocative because they don't fit into any existing paradigm (then they are automatically left in nominative), it is virtually never appropriate to use the vocative form of my name to address me in a language that doesn't have it. Subjectively, it would feel as taunting.
In conclusion, I would say don't use declension in English. Even if you may be perfectly certain what the correct form of the word is, it does not mean it will automatically be appreciated by speakers of either of the languages, and for the reasons outlined above, it's not justified anyway.