Drywall – Do I need a header when non bearing wall

Go ahead and place the drywall over the top edge of the crown moulding. Sounds like the wall was already re-plastered with it in place. If you tried to remove it, you may end up re-doing the walls as well. Even forgetting that, it's almost impossible to remove moulding without sustaining some loss. Without replacement material available, removal with intents of preservation is extremely risky, especially with stain grade material. You don't want to go there.

If your joists were flat, you'd still only need to shim down the first 4 feet or so along the walls, no one will see the resulting slight slope. Keep this in mind when planning your shims. It doesn't have to be perfectly flat, unless that gives you pleasure. You also don't need to shim to completely fill the gap. An 1/8" gap makes a clean shadow line that is not detrimental (assuming it is perfectly uniform) at all. It also makes finishing easier. The taped joints of drywall are normally not perfectly flat, despite the tapered edges. There is a slight bulge which is usually invisible. But against straight moulding, depending on the profile, this bulge may be visible. You might consider grinding down the edge tapers at the board corners against the wall so the joint tape can be completely buried and the yet the joint against the moulding can still be perfectly flat.

I called an engineer and paid for his evaluation of the wall. It's not load bearing. Realized that I left out some basics about the house.

This house is built on a raised foundation with cripple walls attaching the floor joists to the foundation. The metal post is attached to the fireplace foundation (which is independent of the houses foundation). There is no foundation (just 3 feet of empty air) underneath the wall I want to remove. The header is just oddly sized.

The header does not connect to the beam.

From a sheer wall perspective he was not too concerned, given that the total amount of wall I want to remove is pretty minimal compared to the rest of the house.