1. is incorrect. 'ten' is modifying a singular noun.
2. is OK if 'ten paychecks' is interpreted as a unit, e.g., a bundle of ten.
I have every (bundle of) ten paychecks for the last year.
If 'bundle of' is omitted, then the sentence will be interpreted as either awkward ungrammatical.
3. is incorrect for the same reason as 1.
4. is incorrect. The verb should be singular.
Note, if 'ten cars' is interpreted as a unit, then 'Every ten cars' is OK, but since the subject is a singular unit, then the verb should be singular as well:
Every
row of 10 cars
is good. (Subject: row; Verb: is)
5. is incorrect for the same reason as 1. 'ten' modifies a singular noun.
6. is correct for the same reason as 2. 'ten minutes' is a unit of time.
7. is incorrect. 'every' is specific, whereas 'a' is non-specific. Pick one or the other, not both.
a paycheck (det + noun)
every paycheck (adj + noun)
a paycheck pack (det + adj + noun)
every paycheck pack (adj + adj + noun)
a ten-paycheck pack (det + adj + noun)
every ten-paycheck pack (adj + adj + noun)
'every' modifies the noun 'pack'