Glad to hear that he's taking in more milk. While it can be a blessing to have a baby that's so easily contented, it can be quite confusing when it come to feeding! The health assessments at the MCHC will also help to reassure you as the nurses and doctors there are very vigilant and will let you know if you have anything serious that needs to be addressed.
On your mother, are you able to tell her that she's not being helpful to you or your baby? For one thing, she's not respecting your choices and for another, comparing children (and grandchildren) is very unhealthy not just for her, but is negative to everyone involved. I had regular discussions with our parents on both sides to make sure that they understood how we felt about this as well as any other behavior that we felt was not constructive in the upbringing of our children. When stray comments do pop up from time to time, not necessarily from our parents anymore, but from other adults, we address this comments both to the adults as well as to our children who are within hearing now that they're old enough to understand. Where we can be polite, we will deflect them, but we will be sure to make it abundantly clear to our children that the views are not the views of their parents or anyone else who truly loves them. You might want to tackle this (when you have the energy) earlier rather than later.
On the eczema, we use Cetaphil for bathing and Ecolab aqueous cream to moisturize after every wash. We also wash the children's clothes and bedding in detergent and fabric softener that is specific for children. We like Kodomo, but we get our supply from Singapore:
http://www.beautynscent.com.sg/. Another thing that we do use is a bit unusual is bittermelon. We heard it recommended by a confinement lady. Take a small piece, about 1 inch, of the bitterest bittermelon you can find - the small ones used for Indian cooking are PERFECT - and mash with a clean fork or mortar and pestle using a splash of boiling hot water. Allow to cool covered in the refrigerator so that it doesn't evaporate. Then, use a cotton pad to dab onto the rashes. I have absolutely no idea why it works, but it just does, so I thought I should pass the information on. We do keep topic steroids on hand for bad cases, but they have become more rare as we continue being vigilant about addressing any outbreaks as described above early and quickly.
Good luck with everything!