I have a suspicion that you can use all nouns after 'type of' as if they are uncountable nouns regardless of what dictionaries say, e.g. 'a type of pie' instead of 'a type of pies' (if you think 'a type of pie' sounds worse than 'a kind of pie', don't nitpick, it's not the point). Does it mean that, for example, I can say 'a type of missile' instead of 'a type of missiles'? Can the latter be used at all (not just with this specific noun but generally)?
Grammar – ‘One Type of [Noun]’ vs. ‘One Type of [Plural Noun]’
grammarsingular-vs-plural
Related Solutions
I'll give you two other sentences with "is it" and "are there": "How many miles is it to London" (I'm asking about the distance). "How many pubs are there on the way to the park" (I am asking you to count the pubs).
When I say "how many stops is it to the park", I am asking about the distance. It's not a distance in miles, but if you are used to taking the bus, you have a feel for you long it is from one stop to the next, so this is a good way to estimate the distance. On the motorway I could ask "how many exits to our destination", same thing, it is equivalent to the distance.
When I say "how many stops are there to the park", I am asking about the number of stops. I could ask "how many road signs are there to the park", or any other thing I would like to count.
For your first question: The word 'of' is these cases is being used as a partitive; it's used to connect some quantitative description to a group or collective noun, to specify a certain subset. This is also commonly used for counting numbers of a particular set, e.g.:
Six of my students got 100% on the exam.
Four of his friends went to the party.
The linguistics here are beyond me, but in English at least, the group after the noun has to be a specific group, so you cannot say 'all of the people' to mean everyone generally. 'The people' in this phrase will always be interpreted as some specific group of people. Similarly, 'all of the basketball players' will always sound like you are talking about some specific group of basketball players, not basketball players generally. If you said
All of the people are unintelligent
in a conversation without ever specifying what group 'the people' refers to, the listener won't be sure what group of people you're referring to. If I heard that sentence, I wouldn't think it meant all people generally, I would assume that I was missing some of the context.
For your second question: Grammatically, I believe this is the difference between a partitive and a quantitative. When you way
Fifty percent of the people
'of' is again a partitive, so it's functioning to describe a subset of some particular set. In this phrase, 'the people' ought to be interpreted as some particular group of people, just like in 'all of the people'. When you say
Fifty percent of people
'of' functions as a quantitative. Here, the phrase 'fifty percent of' is specifying a quantity: how many people you are referring to. You can't say 'all of people' because 'all' isn't a quantity.
Best Answer
Summary
The question relates to phrases of the structure kind of thing, and asks about accepted forms based on the plural forms of the words within the phrase. As observed, any uncountable noun may not appear in a plural form.
In every case, the original form of the phrase kind of thing is the accepted and expected form for the singular number.
Following are the three possible forms of the phrase that include plural forms of some of the words:
All three forms appear in modern English, as a plural form of kind of thing, and none is unacceptable.
Which, then, to choose, when needing a plural form in speech or writing?
Form (1), though found in respectable works, including those of Shakespeare, appears to have extremely limited use in contemporary speech and writing.
Form (2) is in active use, but appears less commonly than Form (3), which receives the greatest support from contemporary grammar texts and usage guides, and seems preferred by many native speakers.
A safe choice is (3), but (2) deserves better than to be dismissed.
Details
A discussion on the same topic, in the Stack Exchange community English Language & Usage, shows that the topic is complex and controversial, and any overly brief discussion may be inadequate.
(Compared to type, which appears in the original question, kind is a more common word in phrases of the general structure, and will be the preferred example in this discussion.)
Usage
Preference for kinds of thing over kinds of things may be more likely in cases of the following:
Webster's Dictionary of English Usage gives an extended and complex discussion relating to the question (see entry for kind), which suggests that kinds of things may be the wisest and safest choice for contemporary writers, while also explaining that all of the other plural forms have been in widespread use as recently as the twentieth century. The Chicago Manual of Style (16th edition) apparently makes no direct mention of the question, but contains over one hundred instances of the general phrase in the form kinds of things, and none in any other plural form.
Contemporary usage, codified by respected style and usage guides, clearly prefers kinds of things.
Grammar
A Comprehensive Grammar of the English Language (Quirk et al), despite its rigorous treatment of grammar overall, makes only summary mention of the question (see section 5.6, Partitions in respect of quality), suggesting that kinds of things is a preferred pluralization of kind of thing.
Yet, the question appears to be poorly understood grammatically, and lack of agreement over any correct usage from grammatical rules suggests the occurrence of multiple grammatical interpretations.
In particular, ambiguity may arise, within a natural interpretation of the phrases, over the following two competing grammatical parsings:
In case (1), the preceding phrase (in brackets) functions as an adjective that modifies thing, the noun. Then, pluralization of the phrase requires that thing appear in plural form, with no clear requirement for kind. In case (2), the phrase following kind (in brackets) operates on it as a postmodifier, the same as the phrase of blue would do in sea of blue. Then, pluralization of the phrase applies necessarily to kind, but not so to thing.
Considering case (2), applying a plural form to thing may be inappropriate. In sea of blue, blue is an adjective, just as in blue sea. Similarly, because accepted grammar allows tree types, but clearly rejects trees types, the interpretation of the phrase of thing as a postmodifier suggests that tree, moved to the of phrase, follow the same pattern as blue, and retain necessarily the singular form.
Further ambiguity, however, arises because English usage allows either the singular or plural form for a word representing a category.
Consider the following openings of two Wikipedia articles, from the English versions:
If, in kind of thing, the word thing may be considered to represent a category, then a singular or plural form seems agreeable.
One advantage of kinds of thing is that it more closely follows the form required for uncountable nouns. Since kinds of rice is a plural form, and since kinds of rices is not on offer, kinds of potato might be preferred, as the world is filled both with much rice and many potatoes, but the measure of neither is relevant to the counting of their kinds.
Conclusion
The form kind of thing is the singular form of the phrase, whereas kind of things is considered plural, but is no longer common.
The form kinds of thing is an accepted plural form, and may be a suitable one, in many cases, but kinds of things enjoys greater support in contemporary usage and better favor from individuals currently. The latter is a safer choice, but some writers may prefer the former for a particular literary or formal style, or because of certain grammatical considerations.