Both are OK. 1. is common:
A 1.: A salary job. => The job comes with a salary.
A 2.: A salaried job. => The job is salaried. (A salary is given)
A 3.: I got a salaried job. => The job is salaried (A salary is given)
'with cat' means, to be pregnant with a cat (i.e., I am with child means, I am pregnant:
6. I am with cat. (OK, but odd--are you a mad scientist?

)
7. I am with
a Jack. (OK, 'a' means, any Jack)
Wife: Who are you with?
Husband: I'm with a Jack (i.e., I don't want to say which Jack, so I use 'a')
Wife: 'a Jack'? Why won't you say which Jack? Are you trying to hide something from me?
8. I want a dog and cat. (OK. It means, I want them as a pair.)
9. I want dog and a cat? (Not OK. The determiner needs to come first. English is a head first language, so modifiers need to come first; the second modifier can be omitted, as in 10 below.)
10. The dog and
(the) cat are separate.
As for 11. and 12., since you're talking about a specific reader (the person who will read your sentence) use 'the':
11. When I say this,
a reader knows 'the' is omitted right? (Odd)
12. When I say this,
the reader knows 'the' is omitted right? (OK)
13.
This is a dog and a cat. (OK)
14.
This is a dog, and this is a cat. (OK, but use a comma)