The verb must agree with the grammatical subject. Even though the subject in these constructions is a phrase containing two nouns, only one of the nouns is the "head noun", which is the one that matters for subject-verb agreement.
Taking Swan's examples,
This sort of car is...
The head noun is "sort". You can determine this because the predicate will describe the group of cars mentioned in the context, not just a single car; and because "car" cannot be the head noun since it is the object of the preposition "of". "Sort" is singular, so the verb "is" is singular.
These kinds of car(s) are...
Here the parentheses are used to mean that the sentence could say either "car" or "cars", and there's not much difference in meaning between the two:
These kinds of car are...
These kinds of cars are...
Now the sentence describes multiple categories of car, like perhaps two-doors, hatchbacks, and sedans. The head noun is "kinds", the word "car" or "cars" is the object of the preposition "of", and the verb is plural "are" to agree with plural "kinds".
Cars of that type are...
This time "cars" is the head noun, and the prepositional phrase "of that type" clarifies which cars are described. The verb is plural "are" to agree with plural "cars".
Note there's not a lot of difference in meaning between these two sentences, just a somewhat different emphasis:
Cars of that type are sometimes considered unreliable.
That type of car is sometimes considered unreliable.
Swan's other point is a bit tricky.
I'm interested in any new kind of developments...
It's a bit unusual to use singular "kind" with plural "developments" in this way, but it's acceptable and a native speaker might use "kind of developments", especially in speech or informal writing. It could be clearer to use "kind of development", "kinds of development", or "kinds of developments". The three phrases would not have much difference in meaning, except using plural "kinds" does hint that the developments could possibly fall into multiple useful but unspecified categories.
Does this mean you say "These kinds of car are", or you switch the 'are' to 'is' in the case with a singular noun such as 'car' and say "These kinds of car is"?
The verb should agree with the head noun "kinds", not the word "car" which is the object of the preposition "of". A native speaker would rarely say "these kinds of car is...."
"These sort of cars are ..." (...) This structure is often felt to be incorrect, and is usually avoided in a formal style.
Native speakers will sometimes use a sentence like this one, especially in speaking. But notice it breaks the usual rules of grammar: The head noun is singular "sort", and yet it is modified by the plural possessive pronoun "these" and is mismatched with the plural noun "are". This comes from treating the noun phrase "sort of cars" as a plural unit, since it describes more than one car. You might want to be aware of this type of sentence in case you come across it, but I wouldn't recommend using it as a non-native speaker.