Random Function size
Yo are taking only the one-to-one functions, while there are a lot of other functions, to name a few:

00 00
01 00  
10 00  
11 00

00 01
01 01  
10 01  
11 01

00 00
01 00  
10 01  
11 01


as you already mentioned, each table can be represented with $n*2^n$ bits, i.e. strings from the set: $\{0,1\}^{n*2^n}$, each string from this set defines one function from $Func_n$, so the number of such functions is the size of the above set $=2^{n*2^n}$