ALCON,

My wife and I have been engaged in a great debate, and a simple google search has failed to resolve it.  We pose to you this inquiry involving the use of "could have" "should have" and "would have" and whether or not using them alters verb tenses following them.

I did the dishes.  (correct)
I done the dishes. (wrong)

I would have did the dishes. (wrong?)
I would have done the dishes. (right?)

Does the use of would have, could have, or should have alter which is the correct choice to use properly?

The reason why a Google search does not answer your question is because it is way toe specialized and beyond the grasp of Google technology. Google simply sweeps up everything that is out there on the web, and there are no enforced standards for what goes out on the web, minus some stuff about illegal content. You are interested in a specific form of English, Normative English, but there is no way to (a) authoritatively determine whether a web page follows the rules of Normative English or indeed (b) to absolutely determine what the rules of Normative English are. Very much of the content of the web is in non-Normative English, indeed a huge amount is in non-native English, to the point that utterances ostensibly in English are really the result of e.g. a Chinese speaker learning enough English to post a question or comment. There are a number of people trained in Normative English, and through apostolic tradition we know that "I did the dishes" is Normative English", likewise "I have done the dishes". You want to direct your question to them.

One weakly useful Google approach is the literal search, where you put sample pair in quotes and count the hits. "I did the dishes" gets twice as many hits as "I done the dishes". In the case of "I would have did/done the dishes", the number of hits is so low that you learn nothing. With a more generalized search for "I would have did/done", you find that "I would have done" is 7-fold more popular; the most general search for "have done" vs "have did" favors "have done" by about 250-fold. This is a very crude method, because included in the set of hits for "I would have did" are many examples of Normative English grammarians saying "You can't say 'I would have did'".

Linguists are probable very bad people to ask about this, because we are aware that this is a property of Normative English, and not a general property English. Usually if you direct such questions to a Normative English focused audience (such as English Language stack exchange, or any of a number of English grammar fora), the audience will ignore non-Normative dialects and just say "You can't say 'have did'".

I'm not aware of any, and I wouldn't trust it if there was. Counting "words" is an incredibly complicated topic, because it's not clear how to define them. Chinese and similar isolating languages (with few or no affixes) probably have the fewest words, but they still combine to form complex phrases. English is somewhere in the middle. At the other extreme, there are languages with thousands of word forms (even hundreds of thousands for languages with many possibly combining  affixes), and even some languages which have recursive (repeating) morphology and therefore technically infinitely many word forms.

This means "how much vocabulary someone knows" is a tricky question for that reason, even in a single language, and much more so across languages when the rules of the game are different in the different languages. There's very little difference between Chinese, with many small words, including many grammatical particles, and a language with highly complex morphology with very long words even forming a whole sentence. In the end, on average, the parts are often just about the same, whether or not they make up "words".

Furthmore, studies or statistics purportedly measuring vocabulary size are almost always hopelessly biased toward "dictionary words", the sort of thing that would be found on an academic test. And that's simply not a representative way to measure how many words we know, because we know so many other things-- technical jargon and slang are really the same thing, just in different domains; and also consider how many proper names (people, places, brands, etc.) we know. And those things just aren't on the test. A reformulation of your question that might make sense would be to ask whether someone who knows more specifically relevant technical jargon is more productive in the workplace (for example, how many words to they know related to photocopiers?), but that seems superficial to me, at best. Regardless, anyone who needs a word for something either learns it (it's easy to learn a specific word or a specific set of words through usage, regardless of overall vocabulary size), or makes up a word or expression in place of it. We have many such tentative circumlocutions that we use every day when we don't have exact terms for things, and sometimes they become the standard term later anyway.

More abstractly, it is unclear whether we can even define "word" cross-linguistically in any meaningful way. See Haspelmath 2011: https://doi.org/10.1515/flin.2011.002 (draft: https://www.eva.mpg.de/fileadmin/content_files/staff/haspelmt/pdf/WordSegmentation.pdf)

To get around this, you might ask something else, like how many memorized units (whether independent words or affixes) a language has. But that gets into another problem: there are many good arguments for treating some phrases as memorized units too (idioms, as well as the foundational arguments for Construction Grammar). So that doesn't really solve it either.

There may be a useful way to operationalize this question, but it isn't very meaningful on the surface. And even if you found a correlation, there are more complex issues about ideas of linguistic complexity here: is it more efficient to have fewer forms, or to have more forms? It's a trade-off between speaker and listener, being more explicit or not. Complexity is complex, and a controversial topic at this time. However, you might be interested in some of the recent literature about sociolinguistic correlates of linguistic complexity, such as the idea that an influence of having many adult second language learners (as a result of intense contact) tends to simplify languages, so relatively small, relatively isolated languages tend to have more "complex" grammar, by some definition/metric of "complex". McWhorter, Trudgill and others have discussed these issues in detail.