This is a complicated question among chemical educationists- with tons of arguments over arguments. Just search do equilibrium constant have units? I would say the "units" of K are in a grey area (just like in any real world science.). You should read the nice section on Dimensionality of the Equilibrium Constant, and I will just give a background:
Now, after ages, I feel that there are mathematical benefits of sticking to the formalism of unit-less equilibrium constants. For example, you need to take logarithms of K. Quantities in transcendental functions (log, ln, e, sin, cos) cannot have units, just like pH cannot have units. It is then beneficial to keep these quantities dimensionless.
Yes, one simple argument is that we use the concept of activities and that do not have units. However, most of the calculations involving equilibria use concentrations. So the idea is to make the concentration of each species dimensionless by dividing it with unit concentration. Now, whatever the exponents be in the equilibrium constant expression, you don't worry about the units.
Regarding your query regarding rate constants, one may try to use the same argument of using dimensionless concentrations in the rate expressions. There we have logs and exponential functions too. Otherwise your dimensional analysis is correct. In general we can never write the rate expression or determine the order without an experimental support until and unless the reaction is a so-called elementary reaction.