Yes, it is possible to have eczema that doesn't itch. Mine was located on my upper forehead near my hairline and only itched if I was sweating. The cortisone cream would help for a few days but then the rash would come back after stopping. Long term use of cortisone is about the worst thing you can do to your skin so I needed another option. Eczema is such a broad term and can take many different forms but my guess is that you're experiencing some sort of dermatitis that needs to be soothed, not killed by cortisone, BP, etc. I was skeptical about the Neo but it seriously dropped my jaw after the first night of use.
It sounds like you have eczema based on the description of the rash and ability to heal temporarily with cortisone cream. I had the same issue, although in a different spot on my face, and had unbelievable success with Neosporin Eczema Essentials Daily Moisturizing Cream. It's about $12 at Walmart and cleared up my problem within two days. The itch went away almost instantly and the bumps were completely gone (and no longer red) within a week.