[copied from ELU; I'm not sure why advice was given to resubmit the question here. Queries of the correctness of multiple sentences are usually off-topic on ELU, but this question essentially asks about the count and noncount usages of 'origin' / 'origins', a single topic.]
Please look up various existing threads here on ELU, on noncount and count nouns. As CGEL says, it's actually very unhelpful to consider nouns as inherently count (consider the noncount usage of coffee, say:
- "Coffee is my favourite drink")
or noncount (now consider
- "Two coffees, please" and "The two most widely grown coffees are arabica and robusta").
It's usages that are either count or noncount. Many nouns can swing both ways.
And note that a few plural-form noncount usages exist:
- Damages were awarded to the defendant by the court.
- *Three damages were awarded to the defendants by the court. [asterisk showing unacceptable sentence]
With 'origin', the picture is even more complex. Both the singular and plural forms are available, taking the expected agreement (origin is; origins are), but as Longman says, they are usually interchangeable other than as regards verb agreement (unlike say coffee / coffees).
Note also that Longman does say that 'origin' can be used in both count and noncount [(C,U)] ways. The count usage (do not confuse this with the common noncount plural form; we wouldn't say "My 5 ethnic origins are in South-east Asia", so this is a plural-form noncount usage) is rare. I have found an example in a scientific article from Proceedings of the National Academy of Sciences of the United States of America, PNAS:
Two of the three replication origins in each species were located in
the vicinity of a cdc6/orc1 replication initiation gene ...
As might be expected, the count usage focuses more precisely on an origin / source / birthplace than the more inclusive, general noncount usage. But the broader, more general (perhaps 'fuzzier' is too pejorative) sense is what is usually the more convenient one.
Here, examples 1,4,5 and 8 are ungrammatical. 'Interchangeable / synonymous' yes ... but the required verb-form may need adjusting.
2 and 3 seem conceptually muddled (origin/origins are usually non-sentient; 'originator' would be somewhat better).
While 6 and 7 are both acceptable, 7 is more commonly found according to Google Ngrams. However, the plural-form variant might well be preferred to hint strongly at complex, detailed beginnings. And note that 'beginning/s' behaves rather similarly.
The Cambridge English Dictionary gives more detail on when the plural- or singular-form noncount usage might be (or must be) preferred:
(a) It's a book about the origin/s of the universe.
either totally acceptable and idiomatic.
(b) Her unhappy childhood was the origin of her problems later in
the plural form would not co-exist happily with 'was', required by the subject.
(c) He is of North African origin. / (d) What is your country of
'of origin', not 'of origins'.
(e) We begin our dip into local history by examining the town's
arguably the more idiomatic, the plural-form noncount usage strongly connoting (at least) a complex, multifaceted early history
(f) The Easter egg has both pagan and Christian origins.
certainly needs the plural form, but I'd again argue against this being regarded as a count usage. 'The Easter egg has several origins, both pagan and Christian' doesn't work. 'Origins' is here more unitary, an agglomeration of strands rather than strands considered totally separately. Note that 'The Easter egg has origins both pagan and Christian' is perfectly grammatical if a little rarefied.