You have two firsts, an also and a final question. Let's hope your mechanical skill exceeds your writing.

Starting with the fundamentals. The fork most definitely needs a crown race. These are usually a press fit and require a slide hammer (a faced pipe) to press them on. Some cartridge bearing headsets don't take a crown race per se, since that's part of the bearing, and instead use a split cone that works similarly to the centering cone at the top.

If it's a press-in headset (yours looks more like an integrated headset. Pressed-in cups have cylindrical extensions about 1/2" long to fit into the head tube) you need a tool to press it in.

Unless you are 110% sure (as an expert, not an amateur) of what you're doing, and why DO NOT SAND OR OTHERWISE MODIFY CARBON PARTS.

Given where you are, and your questions, I'm uncomfortable trying to help beyond this. You're clearly unfamiliar with headsets in general, and I don't want to be the one that gets you in deeper trouble. I strongly suggest you see a local mechanic, either at a dealer, or better yet at a bike co-op before you cause irreversible damage to the frame or fork.

Just a few threads away, is one from someone who destroyed his brand new warranty fork because he was working above his skill or knowledge level. There's no sin in not knowing, but there is in not being willing to say "I don't know enough and should leave it so someone who does"