Are there known to exist functions with the following direct-sum property?
Well, that is false: it is possible to evaluate M copies of ANY f using only O(N(M+2^N)) gates which can be much less than M*exp(N) (in fact, you get linear amortized complexity for exponential M).  I don't remember a reference, but I think that it can go something like the following: 

First add 2^N fictitious inputs which are just constants 0...2^N-1 and now denote the i'th N-bit input by xi (so for i&lt;=2^N we have xi=i, and for 2^N &lt; i &lt;=2^N+M we have the original inputs).  Now we create a triplet for each of the M+2^N inputs: (i,xi,fi) where fi is f(i) for the first 2^N inputs (a constant that is hardwired into the circuit) and fi="*" otherwise.  Now we sort the triplets (i,xi,fi) according to the key xi, and let the j'th triplet be (i_j,x_j,f_j) from this we compute a triplet (i_j,x_j,g_j) by letting g_j be f_j if f_j is not a "*" and let g_j be g_(j-1) otherwise.  Now sort the new triplets back according to the key i_j, and you got the correct answers at the correct places.