The woman(1) and its(1) dog(2)* went shopping, and it(1) bought a pair of new knickers.
Now, isn't that much more neat and tidy and totally understandable? Why can't all English be as clear and easy to follow, hmmm?

But CapK, your reconstuction only works because of the specific facts of that particular sentence. It wouldn't work, for example, with:
The woman(1) and its husband went shopping, and it bought a pair of new pants.
Whose pants would they be: a pair of "his" pants, or "hers" pants? Whichever you say, how would you alter the sentence if you intended to say the opposite? Yes, it can be done, but only with awkwardness.