For example, in the sentence: 'that book weighs six kilos'...what is the word class of 'six'? I know some grammars have 'numerals' as a word class, but if you don't have that, what is it? an Adjective?
It depends on your ideology of word class, and probably on the language (depending on your ideology of word class). In a number of Bantu languages, they constitute an autonomous word class. They have a unique pattern of class agreement making them distinct from adjectives, quantifiers, and determiners; they have special morphosyntactic rules of combination (do not combine with certain definitizing prefixes, the so-called augment), have special word order (after adjectives, before possessives), and also have distinct patterns of reduplication. The morphological diagnostics actually justify further subdividing numerals into "1" vs. higher numerals.
You say "if you don't have that...", but the question is, why wouldn't you have it? That is, to simply stipulate that a language can't have a word class "numeral" is arbitrary and problematic for languages where there is good evidence that they are distinct from other word classes. This doesn't mean that every language has to treat numerals as a distinct word class, so it is possible that there are languages where there is absolutely nothing that distinguishes numerals from adjectives, for example, though I can't think of an example. If your theory of word classes isn't based on an a priori mapping between semantics and word class, then I think the answer would be something like "whatever the facts of the language require".
It seems in English, numbers can be adjectives, determiners, and nouns.
(Though I feel the nouns are really just adjectives that have an implied noun.)
Types of numbers
Ordinals: first, second...
Cardinals: one, two...
Ordinals seem comparable to superlatives
"I am the very first Uber driver." is similar to "I am the very best Uber driver."
"For my third trick..." is similar to "For my next trick."
"In my sixth year" is similar to "In my final year..."
Cardinals seem to work here
"Six years ago..." is similar to "Many years ago..."
"Here, have six dollars." is similar to "Here, have some dollars."
"One week had passed." is similar to "A week had passed."
"He is my second." ... I feel a noun is implied via context "He is my second son." just like "He is my favourite." implies in context something like "He is my favourite character."
"You are the chosen four"...implies "You are the chosen four winners," similar to "You are the chosen few." "You are the few winners that we have chosen."
"You are the one [person in the world] for me."
"I'll take those two [things], thanks."
"I am the very one you seek." similar to "I am the very man you seek."