First, I want to respond to the comment on the question, that was added by OP. The other answer does not say that "when contraction is possible, it is strongly favored over not contraction" Rather, the quote from the Longman Grammar is given in support of the very specific context of whether:
I've not
is preferable to
I haven't
Use of contractions is not preferred. It may even be discouraged in written English. It depends on the audience and the tone of the writing; however, single contractions are very common in spoken English.
No, there isn't a rule about order of precedence in contractions.
Yes, usage does depend on the particular situation and meaning to be conveyed.
Double contractions are used colloquially. They are much more common in the language as spoken, rather than as written. This is true for both American and British English. Here's a list of double contractions via BBC Learning English.
Here's BBC Learning English emphasizing the importance of NOT using double contractions, in its Stop Saying That series!
So a double contraction is when we make three words into something
shorter in spoken English. I would have becomes I'd've and could not
have becomes couldn't've. They're quite common in spoken English but
not used very often in written English. They might be used for
informal communication, but Helen recommends you don't use them for a
job application!
Contraction vs Elision
The question title mentions elision. I want to clarify the difference between the two words. Contraction is a general term for combining two words to make one short word. Elision is a specific term for omitting sounds or syllables, and replacing them with an apostrophe.
Elision happens a lot at the ends of words. For example, shortening
“fallen” to “fall’n” and “tired” to “tir’d.” Elision occurs in the
middle of words, as well. For example, changing “every” to “ev’ry” or
“never” to “ne’r.” Elision is used to deliberately maintain a rhyming
scheme, say in iambic pentameter, i.e. as a literary device.
There isn't a lot of elision in modern English communication, written or spoken, although there was in pre-21st century English literature and poetry. There still is, sometimes, depending on the prose writer or poet's preference regarding tone.
