It can be zero and does not need to be an integer. You'd have to know the mechanism of your reaction.
And so we use the initial rate, where we have only reactants present, and no products. There are dimensions to k and that be determined with simple dimensional analysis of the particular rate law. A new, especially reactive segment of the reactant, called the saddle domain, is introduced, and the rate constant is factored: k.
So two, all right, so two to what power X, two to what power X is equal to one? For nearly all forward, irreversible reactions, the rate is proportional to the product of the concentrations of only the reactants, each raised to an exponent. And so those would also be your units for K. K would be in molar per second.
If you increase the concentration of your reactants, you increase the rate of your reaction.