Classifiers are mostly orthogonal to the number system, and they're really no different from e.g. English using "a herd of cattle" or "a murder of crows" to redundantly encode the kind of thing being talked about. I'm actually not aware of any language that doesn't have different units of measurement for different objects, it's just that some languages effectively make it mandatory to use them. (E.g. in Chinese you could just always say 一个 to mean "one" but it sounds strange when there's a specific form that's more common.)
Yes, they are different from English. In English you say "one herd of cattle", "two herds of cattle", and so on. You just need to know a plural of "herd" and the regular numbers.
In Japanese the numbers themselves depend on the counter. You can't say "one day", "two days", "three days". You can only say "one day (ichinichi)", "foobar (futsuka)", "bazbar (mikka)", and so on. And this counting system is used pretty much only for days. Except similar words to "foo" and "baz" here are also used to count some things. But not the others.
Also the rules of combining a numeral and the counter are very inconsistent. Here's my implementation of Japanese counters, suffices to say the English version would be like 5 lines of code:
The completely different number words used for some things are actually not related to the counter, although it may appear like that because the written form uses the Chinese-style number + counter system. The spoken words are holdouts from ancient Japanese (before Chinese become influential), which TFA discusses separately.
The situation is similar to Latin or Greek numerals in English, which are also applied pretty inconsistently. E.g. duplicate, triplicate, quadruple, ... but pair, triple, quadruple, ...
In general, all languages are riddled with exceptions, ambiguities, contradictions and archaic forms, but it's easier to notice them when you're learning the language later in life and have the "logical" structures of your mother-tongue to compare them to.
No way. It is absolutely different in practice because of Japanese counter words.
A counter word is a counting prefix (different from the abstract number word) plus different suffixes for a variety of things. This object is long and thin, so it has one suffix. This object is a drop of liquid, so it has another suffix. This is a small animal, so it gets another suffix. The counting words for use with these suffixes are different from abstract numbers.
Frankly, the system is bananas.
Abstract numbers in Japanese are very simple and logical, but the overall counting system of e.g. English is objectively simpler, and I don't think you can disregard counting when examining the number system.
In English, it would be equally correct to say "10 cows" or "a herd of cattle." Literally translating the first (without counting words) in Japanese would be incorrect.
The example was poorly chosen, but obligatory measure words are very much a feature of English; compare "a pair of pants" / "a pair of scissors" / "a pair of glasses" / "a loaf of bread". Note that all of those "pairs" refer to a single object.
Moreover, Japanese has TWO parallel number systems (as does Korean). The Sinitic system, enumerated here, is used for naming numbers, counting with most counter words, and counting large quantities, whereas the native Japanese system is used with certain common counter words.
The native Japanese numerals are themselves variable: I have listed the forms they assume when paired with the つ counter. For example, two days is rendered as ふつか and six days as むいか.
