These look like bugs to me. MixedRadixQuantity is a constructor that creates Quantity expressions of the form: Quantity[MixedRadix[...],MixedRadix[...]]

It seems to be failing to convert many of these, but you can re-write them in this form and they should work: Quantity[MixedRadix[1, 2], MixedRadix["Meters"^2, "Centimeters"^2]]

Note that "Meters/Seconds" isn't a canonical unit form(it should be "Meters"/"Seconds"):

In[14]:= KnownUnitQ["Meters/Seconds"]
Out[14]= False

In[15]:= KnownUnitQ["Meters"/"Seconds"]
Out[15]= True

This parses automatically in Quantity, but not in MixedRadixQuantity(another probable bug); Also note that while some of these exponents will typeset, they don't work in standard Quantity expressions:

In[18]:= Quantity[1, "Meters"^E] // UnitConvert
Out[18]= UnitConvert[Quantity[1, ("Meters")^E]]

In[19]:= Quantity[1, "Meters"^E] // QuantityQ
Out[19]= False

(I will confess that I'm not quite sure what you're trying to do with these however)