Re: Annealing for first time.
Both. The neck and shoulders could become too soft, which would show up in poor neck tension or damaging the case when you resize. Over-annealing can cause a decrease in accuracy until the neck hardness is restored slightly. The lack of neck tension can also cause the bullet to setback when fed into the rifle, which could be a potential safety concern.
If you anneal way too much or incorrectly, you can soften the area just above the web of the case, which can cause case head separation and, therefore, case failure. I've only heard of this happening when guys stick their brass in the oven thinking they can do all their cases at once.
Edit: I intentionally over-annealed quite a few cases when I was starting out, just as experimentation. Since I've started doing cases that would actually be fired, I haven't over-annealed any. I've probably done about 600 cases now. I use the darkroom method, as judging by color will change depending on the cleanliness of the case, the specific propane being used, and the type of brass used in the case itself. If I clean LC brass with stainless media, then anneal it after about a week, it won't develop the blue tint at all. It'll stay golden red around the neck and won't change colors at all below the shoulder.