We have a washing machine just for mops / rags and I certainly wouldn't recommend washing either with regular laundry. At a push I'd use the same machine but I'd be rinsing out after the mops prior to washing our family laundry.
Never had a problem washing any mop head out. We use socket and Kentucky and both wash with no problems (apart from the thin twin Kentucky heads, they tangle really badly so steer clear).
Usually purchase from LMB supplies.