Should I destlangize the word, or leave it in its “native” form?
In natural languages, borrowed words are almost always "destlang-ized" to some degree, but it won't necessarily always to the same degree. Even within the same language, often some borrowed words will be more integrated into the language than others. The more recently a word was borrowed, the more speakers of the language will treat it as a "foreign" word and be willing to make exceptions for it, but as its use becomes more and more common, speakers will inevitably begin to treat it as a native word if it doesn't fall out of use.
Does the borrowed word take on the conjugation/declension pattern of destlang?
Sometimes, sometimes not. There are examples of both in natlangs.
The English verb "to google" has been borrowed into many languages, and it seems to typically conform to the conjugation patterns of those languages. Even in Japanese, a language with a famously closed class of verbs, borrowed "to google" as a verb ググる(it may have become a verb after being borrowed as a noun, but either way, it's quite remarkable).
Yet it's also possible for exceptions to be made. This happens with some English noun borrowings (most English speakers won't try to pluralize edamame or Pokémon with English plural morphology), but not with others (many English speakers will pluralize zucchini as zucchinis, and you'll pretty much never heard zucchino).
In my experience, verbs are more likely to take on the native conjugation patterns than nouns are native declension patterns, but I don't have any data to back that up cross-linguistically. A borrowed word is definitely more likely to be conjugated or declined like a native one the longer it has been in the language (the pedantry about pluralizing Latin borrowings in English is an exception to this, but that's more because of the social prestige associated with having studied Latin).
Does the borrowed word get re-spelled to conform to destlang’s orthographical conventions?
This really is a case-by-case thing. If the language uses a completely different writing system, almost definitely, but sometimes the orthographical conventions won't be the same as native words. Japanese uses katakana for borrowings, for instance, and English borrowings from Japanese and Chinese tend to follow whatever romanization scheme was most popular when they were borrowed, even if the pronunciation is pretty opaque. For instance, the Chinese word 道 was borrowed as "Tao" in English based on older Chinese romanizations, even though the Chinese word now romanized as dào and the English derived word "Taoism" is more often pronounced /ˈdaʊ.ɪzəm/ in English (though a spelling pronunciation /ˈtaʊ.ɪzəm/ has arisen due to this).
Does the borrowed word’s pronunciation get changed to conform to destlang’s phonemic conventions (e.g., elimination of consonant blends, all syllables must end in a vowel, etc.)?
Almost always yes. While sometimes educated speakers or upper-class speakers will try to pronounce a borrowed word with its "original pronunciation" rather than its borrowed one ("gyro" is a good example of this), even those pronunciations are almost always somewhat changed to fit the destlang's phonology better, and often they aren't even closer to the pronunciation in the sourcelang. How exactly the word is altered to fit the destlang's phonology depends on the particulars of the borrowing situation, however.
What sort of methods exist for choosing which words to borrow? What causes a “native” word to fall out of use in favor of a borrowing, or, contrariwise, why would both the “native” word and the borrowed word stay in use?
I've combined these questions because I feel there's a lot of overlap in the answers. Often, words are borrowed to fill a semantic gap in the destlang, which is why words for new technologies are so often borrowed (think of how many languages borrowed the word "television"!)
Borrowings are also often used to refer to the "versions" of certain things from the part of the world that speaks that language -- consider how anime, a generic word for all animation in Japanese, was borrowed into English as a word for a style of Japanese animation, or how English borrowed "chai" from the Hindi/Urdu word for tea, चाय (cāy) / چای (ćāy) to refer to tea with certain spices based on Indian recipes.
Also, sometimes words are borrowed because they carry some social value seen attractive by the speakers of destlang. Maybe destlang speakers think sourcelang sounds refined and sophisticated, and so upper-class destlang speakers borrow words from sourcelang to sound fancy. Of course, this sort of situation depends on a lot of social and sociopolitical factors wherever the languages are spoken.