Nyati,
This is probably a disappointing answer, but I believe it to be correct:
1) The lack of friction modifiers will NOT hurt anything in a lawn mower, weed whacker, power washer, etc. At MOST, you might experience a slight increase in fuel consumption if you do not have friction modifiers. Practically, this difference is probably not measurable outside lab conditions, and is not that big a deal. BUT: We're oil geeks here, and if your equipment doesn't have wet clutches, why not use an oil that maximizes fuel economy?
2) To the best of my knowledge, you cannot tell from a simple UOA whether or not an oil has friction modifiers in it.
Amsoil 4-stroke oil is an excellent choice, your equipment will last forever if it is used and other maintenance is performed as it should be. But the Z-Rod oil MIGHT perform slightly better in terms of fuel consumption, although in normal conditions, the difference is probably not significantly noticable.