In declarative statements I’ve gathered that saying just 「これ新聞だ」for example would be too forward or a bit aggressive. So could you just say 「これ新聞」or even just 「新聞」for informal sentences?
I understand that い adjectives can be left on their own but I’m just not sure how to write informal sentences with な adjectives and nouns and not using です.
I wouldn't say using ~だ is "too aggressive", though it does have an aggressive feel to me. Especially if you are talking in an informal setting I wouldn't be too concerned with it. (It was formal, you would probably be using 〜です anyway)
I think you can often omit the だ for a softer feel, however in practice ~だね or ~だよ is pretty common, at least in informal conversation.
adding だ without a particle ending as Locksleyu mentioned just seems kind of pointless in informal cases (to my ear). You would typically only add it in combination with a particle like ね or よ. And, instead in informal situations (as you mentioned) you would simply omit it.
Of course this is only in spoken Japanese that typically has a "looser" feel from a grammatical standpoint and is based more on the feeling. And, adding だ where not needed in spoken Japanese does indeed "feel" strange and would probably seem strange to the listener, as you mentioned.
