Is the DES F-function injective for a given subkey? Is it possible for two different half-block inputs to the DES F-function, with the same round subkey, to produce the same output?  

That is, if we feed two different half-block inputs $b_1$ and $b_2$ through the expansion stage, XOR them with a constant subkey $K$, and then pass the expanded and XORed inputs through the S-boxes, is it possible for the outputs $o_1 = F(b_1, K)$ and $o_2 = F(b_2,K)$ to be equal?
Yes, it can; within the DES round function, two different 'right side' inputs can, after the sboxes, come up with the same value to xor into the 'left side'.

This was a deliberate decision by the DES designers, who thought that this was an important property.  I don't know their reasoning about why they thought it was important.