I'm not aware of dust mites causing eczema, more like headaches/ nasal problems but I could be wrong. Maybe your doc could rule this out?
What I would check is some of the ingredients in the things coming into contact with their skin, especially:
SLS - harsh cleanser in practically every shampoo or bubblebath, very drying.
Perfume - increasing numbers of folk becoming allergic/ sensitive to, in practically everything you can buy.
I'd defo do an extra rinse in wash machine, I did that for my son until a short while ago.
Re nappies - give them a few minutes without the nappy to ensure skin dry before applying cream or nappy. I just used water and cotton wool initially to avoid alcohol in wipes and dried with a reusable wipe. Again, most nappies have ingredients you wouldn't think it did so it pays to read the label.
We don't use bubble bath, just use a shampoo once a week on his hair but rinse it with water at other times.
I can recommend A'kin unscented range which has no perfume or sls and really gentle on skin. If you're on a budget you could try Urtekram no perfume range but I much prefer the fab lather of Akin!
Also, Bio-D does an excellent range of laundry products for sensitive skins or you could try soapods which come from a tree so have no additives whatsoever.
Have a look here: www.eco-essentials.co.uk
BTW MIL needs to be given a care plan for your kids, much like one you would give to a nursey. She should really respect your views and you are the boss when it comes to your own kids.
I hope you find something helpful.