Montgomery Algorithm
The confusion comes from the choice of representation. I'd a quick look to the referenced paper, where the autors use a 2-radix representation. Then you shoud initialise $e=\frac{m+15}{w}$ instead of $e=\frac{m+1}{w}$ as you use a 16 bit adder!
The best is to read again the seminal paper of P.L. Montgomery:http://web.itu.edu.tr/~orssi/dersler/cryptography/Montgomery.pdf

ANd this remak explain the additionnal word set to ZERO in the modulus.

The method introduced by Montgomery is clever and require a change or representation as explained in the paper. the method allows also to overlap mul and ReduC process.
I confirm also than C represent the carry which require 2 bits to be holded. The interest compared to the classical method, is that carry propagation is moving from right to left, exactly of the direction of scanning operand. This is the reason why the Mult-ReduC can be overlapped, and which is of great importance in hardware design.