Last time one did any sort of research (ok, skimming "Google" for articles", physicans and others, including consumer groups have found that enzymes in detergents do not cause skin problems. However that does not rule out the odd person that might have an allergic reaction. Such reactions could also be from other chemicals in the detergent such as scent, or residue from incomplete rinsing.
If enzymes in detergents caused so much skin problems, you would see much higher numbers of said problems, at least in Western Europe and the United States, where by and large most laundry products contain advanced enzyme cocktails. Indeed outside of the UK, "non-bio" detergents for the most part are limited to certian MOL and BOL brands. It is worth noting that even some "Green" detergents add enzymes, which somewhat acknowledges that cleaning modern laundry without can be difficult, especially at today's lower wash temperatures.
As for grease/oil stains:
Problem with these stains is often even after the stain is gone, an oily ring will remain on the fabric, especially darker colours.
One might wish to try sponging with a solvent based spot remover or dry cleaning fluid, then working in either a good liquid detergent or laundry pre-treater, allowing it to sit, then rinse, then wash as normal. If going this route, you want to make sure anything remotely flammable is washed out before putting item in the washing machine.
Have a few cartons of a stain/spot pre-treater by Ecolab,which in addition to the normal enzyme cocktail, has some solvents as well for dealing with oils. Works pretty well, just is rather nasty stuff and must be kept off bare skin.
L.
