Sorry.No.You must have an FFL transfer from Ohio to NJ.
Not if you buy at the CMP Store. I believe Ohio only allows sales to contiguous states. CMP does not operate under Ohio law but under regulations established by their charter from Congress.
You do need all the docs required by CMP age, citizenship, club affliation, marksmanship training, AND your NJ Firearms ID Card. Check the CMP website for requirements.
CMP will not sell you a M1 carbine as that is not legal to own in NJ. If you meet the requirements they will sell you any M1, 03, 1917, or other rifle as long as it is legal where you live.
If you go to the store and are qualified to purchase you walk out of the store with it. The only caveat about going directly to the store is what they have on hand varies and you may not find what you want. I was passing by the store on a business trip and did bring home a great Field grade M1.
They do the NICs at the CMP Store (no extra charge). No need to use a FFL if you're buying from them.