Sediment type filters won’t have a problem with (harmful) bacteria but charcoal/carbon filters can and do. That’s why they all come with a warning that they are not supposed to be used on watr of unknown microbiological content.
Just about any water treatment dealer can provide hot water filter housings and hot water water softeners.
My money is on the rust stains being caused by the washer tub rusting somewhere that the clothes are in contact with. DOnt leave the clothes in the machine after they are finished washing and inspect the inside of the tub with a flashlitgh for any brakes or chips in the tub surface etc..
Quality Water Associates