How do coefficients affect the rate law?
Do you mean stoichiometric coefficients? Let's take a classic example.
Each coefficient is worked into the rate of appearance/disappearance like so:
So, we get:
Therefore, the stoichiometric coefficients do not affect how the rate law is written, but they do affect the value of the rate constant
Also, the reaction order does not correspond to the stoichiometric coefficients; it's only a coincidence here. The reaction order only says how many molecules per mole participate in the reaction (molecularity is usually a good indication), not how many moles of each molecule there are (stoichiometric coefficients).