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I've been told that compound contractions like couldn't've and I'd've are proper grammar. Are they?
Internationally, compound contractions are not regarded as correct grammar. Even simple contractions should be avoided, if possible, in a formal or academic context.
High school English teaching in my country, 'English for foreigners' courses, as well as the exam boards which provide English proficiency certifications (Cambridge ESOL) mark them as incorrect. I'm talking mainly about British English here, as it is the more common form of 'language import' in Europe. I don't know about TOEFL or other AE testing institutions.
In the UK (at least in southern England) however, you might find the issue to be a bit more complicated. Double contractions are very common in spoken language, and I have occasionally encountered them on flyers, posters or informal letters. On the other hand, exam boards like OCR and AQA won't accept them as correct, when doing A-levels exams.
So you could say as a thumb rule you should avoid them if you are writing prose in a formal setting. If you are writing for yourself, to friends or family, you might use it. You can also use it within the quotation marks, when writing direct speech of a common person.