Re: 1 month old constantly pooping?!?!?
Could just be his normal BUT the diaper rash is cause for concern. What are you using for wipes/solution? He may not finish pooping because it hurts because his butt is raw.
Try switching to just water to rinse his little bum rather than wiping with any kind of solution. If that doesn't help have you looked at what a picture of a yeast rash looks like? If it is yeast, you will need to treat him as well as the diapers (diapers can be washed with large quantities of vinegar or grapefruitseed extract). Also consider a topical protectant, desitin, balmex, etc (not diaper safe) or a diaper safe balm OR coconut oil (diaper safe). Diaper free time is a GOOD thing to help with healing.
Also consider whether the diapers are making it worse. Some babies are sensitive to detergents, others synthetic fabrics and still others have a problem with moisture and do better with synthetics that wick instead.
Finally, does he have a red ring around his anus? That can be a sign of a food sensitivity. If that is the case, look into an elimination diet, or food diary. The elimination diet will clear up the diaper rash and constant pooping (if the problem is food related).