When a word has both English and 'Latin' plurals, which style should I use?
In general, you can look to overall usage to get an idea of which to use. Searches of COHA, COCA, and the Google Books Ngram Viewer are great sources of data for that. However, one thing to consider is that for many of these words, I would personally use both of the plurals in different situations. So consider using a context search on COCA for perspective when doing research.
For example, I would speak of indices of economic decline, but of tracking major market indexes. Similarly, array indices but database indexes.
I would use minima in a mathematical context, but I would never speak of needing to carry cash because all the restaurants have $10 minima for using credit cards. I would speak of “$10 minimums”
I would refer to a table of formulae in a math textbook but talk disparagingly of those looking for “magic formulas and shortcuts”.
A doctor who has performed many appendectomies I would say has removed many appendixes, but the last third of my calculus textbook consisted of nothing of appendices containing tables of integrals and values of trigonometric functions.
A very short, preliminary answer: it depends on the noun. Some nouns should always take the Latin plural, some can take either, and others should always have the English plural. Even in academic writing, not all Latin forms would sounds proper. If I have time I will look up examples and edit them in.
As an actual classicist, I'd argue for the English plural in all but a few cases. Unless you are absolutely, completely sure you know the correct classical plural, or the classical plural is the normal plural, use the English plural. Using the classical plural may have a nice ring to it, but if you get it wrong it's so, so wrong.
Most generally, American speakers tend to use the English-style plural, while British (and related) speakers favor the Latin-style. Of course, there are those words whose Latin-style plural forms are non-negotiable on either side of the pond, such as bacteria and criteria. And there are also those whose English-style plurals are universally preferred, e.g. moratoriums, apexes, etc.
As Cerberus said, usage differs between nouns. Unfortunately, there are no simple rules that are completely effective at predicting which nouns tend to take which kind of plurals.
The use of one kind of plural versus the other also may depend on context, but the way you've put it is not quite accurate ("In technical language, generally, Latin-style is the only proper form of Latin plurals"). There are many situations in technical language where English-style plurals are correct.
Some notes on specific nouns or categories of nouns:
For crisis and axis, people rarely use regularly formed English-style plurals ending in -sises/-xises. You'll almost always see crises and axes instead. Other nouns derived from Greek that end in unstressed -sis also tend to use the Latin-style plural in -ses to the exclusion of the English-style plural, such as basis, thesis, hypothesis, neurosis.
Nouns ending in -or have plurals ending in -ors in English, not in -ores as in Latin. Thus, we say and write professors, vectors, operators, sectors, etc. (These nouns have various origins; not all English nouns ending in -or are words in Latin.)
Latin-style plurals are extremely rare for the following nouns (this is not a comprehensive list): simile, rationale, specimen, omen, regimen, ratio.
As mentioned in Andreas Blass's answer, some words, such as antenna, allegedly are or should be used with different plurals for different senses.