As a professional linguist, I have some/ many opinions about the discussion here, but a limited amount of time forbids me from going into all of it. I will choose one or two to respond to.
(Note that, just because I am a pro doesn't mean that I have all of the answers, or that everything I say is necessarily correct. This is obvious, since no two linguists have the same opinions, but I just wanted to put it out there. I do have reasons to support my beliefs, and I have put in a lot of time thinking about these and similar issues.)
Question:
what type of part of speech is "what"? How about "how"? "That", etc.?
In intro linguistics, my prof claimed that this was a zone of on-going controversy.
The problem is that each word you mentioned is actually several "words".
For instance, take the word "that". It can be used in several different ways:
1) as a demonstrative pronoun : I like that.
2) as a demonstrative determiner/adjective: I was given that shirt by my Uncle.
3) as a complementizer: He seems happy that you came.
4) as a relativizer: She’s the woman that I told you about.
All four of these realizations are grammatically distinct.
Not only that, but they are actually pronounced differently!
Berkenfield (2001) measured people saying these and found that the vowel is both longer and higher (F1) at the top of this list, and progressively shorter and lower as you move to the bottom.
So there is solid evidence that people are thinking of these four uses of "that" as four distinct words.
I don't think that there is a great deal of room for controversy here, though; describing the facts of language as it is used, we can come up with a list like the one I have above. If you want to argue about how to label each use of "that", you are welcome to (but I won't be taking part in that discussion, which is generally very boring to me). As for how it is used, however, the answer is clear.
There are many other points I could address in this thread, but I will state this with a brief explanation: I, and thousands of other linguists, do not accept Chomsky's claims about Universal Grammar or a Language Acquisition Device. To us, it is simply bad science to make such claims. Saying that there is a "black box" in the brain, ready to acquire language, is not a real hypothesis, but rather a last resort. If all other hypotheses are tested and found to be unlikely, we may have to resort to claims of innateness, but it is better to explain/motivate things based on other facts. In fact, there are many, many aspects of language that can be motivated by biological, neurological, socio-anthropological, and historical facts.
For example, why do the many languages with 5 vowels all have essentially the same 5 vowels (a,e,i,o,u)? Because Universal Grammar specifies that languages with 5 vowels must use these ones? Ridiculous! Better: because they are located at maximally distinct regions within vowel space (the oral tract), which in turn maximizes both ease of production and ease of comprehension.
Why do all languages have a word for "water"? Because Universal Grammar demands such a word? Very silly. Better: because all humans need to drink water regularly, and the communication demands of this real-world behavior result in all languages borrowing, creating, or inheriting a word for "water". Simple.
To motivate language using what we already know about humans and the world is far preferable to hypothesizing that language is innate. The residue, after all potential explanations have been exhausted, can be said to be innate, but to do so before that seems really silly.
A movement sprung up in the 70's and 80's to reject Chomsky's version of linguistics, and that movement has grown into multiple, established schools of thought with thousands of practicing linguists supporting them.
Chomskian linguistics became the dogma for a few decades there, but this is no longer the case. The field is moving closer and closer to an even split between Chomskian and non-Chomskian approaches.
One classic book arguing against Chomskian linguistics is George Lakoff's "Women, Fire, and Dangerous Things" (1987). Though Lakoff was wrong about a good number of things, he got the basic argument right, in my opinion, and laid it out in detail over the course of about 15 chapters.
There are simpler and more profound arguments against UG and LAD, however, as I have described here, and if anyone is still interested, I can go into more detail.