Units continued

Multiply a meter value by a meter value and you get square meters.
Multiply it again by a meter value and you get cubic meters (add a dimension).

This indeed has a relation with the operation you did.
When you divide square meters with meter, you get back a meter value.
There is indeed some rules with the units.

I remember back in school having to deal with kg^2.s^2/m^-3.

Look at the MKSA system and the expression of the common units in international units (Ohm for example).

Anyway, you are right for the mix of the units and the values, but making operations values with units change the nature of the object you manipulated. Think of it like an attribute of the value (like in a class).
ex:
4 meters / 2 meters = 2 (2 nothing, a pure numeric value).
4 meters / 2 seconds = 2 m/s (meter per second or m.s^-1)