Explanation of the dog
I've seen people on this board make references about the dog or speculate on the meaning behind the dog or what it the symbolism behind it is. But is seems to me nobody really understands it, so here's my take on it.
The film is, simply put, about faith. What we see during the film is stalker's obvious belief in the Zone and the Room. It's what he lives for. To him life is in the Zone. He doesn't care if he ends up in prison again, because life outside of the Zone is the same as prison to him. This is why we see the scenes in the Zone in colour and outside of the Zone in black and white.
When Stalker takes Writer and Professor into the Zone we see two characters very different from him, and from each other. Writer and Professor get into an argument in the scene where the three of them are laying in the grass to rest for a while. While Writer and Professor are having this argument (or after) we see Stalker having some sort of a dream. It is here that the dog first appears (in his mind). The dog is connected to the argument (more specific: the beliefs) of Writer and Professor.
Later in the film we see the three of them standing before the Room, and Writer and Professor deciding not to enter the Room. They make it clear they do not believe in the (powers of) the Room. Before this scene we see the dog again, following them. After they leave the Zone it appears the dog has followed them out of the Zone.
At the bar the wife of Stalker asks the men if they are interested in taking the dog home. Writer answers: 'I already have 5 of them at home'. At this point you should realise that the dog is something of Writer and Professor that instills itself in Stalker. More specifically it is a doubt in his belief/faith that Writer and Professor instill in him. It first appears during the argument which irritates Stalker. It creeps up on him during that argument in his dream. Writer saying he already has 5 of them at home is like saying: I don't believe in the Zone at all (5 is more than 1, Stalker now has a little of Writer in him).
Back at home we see the wife giving the dog some milk. I am not sure but this may indicate that the wife will later feed these doubts, because it would be for the best (remember she doesn't want him to leave to go to the Zone at the beginning of the film). Here we also see the ending, with the daughter moving objects with her telekinetic powers. This is (as many have rightly stated) the proof that the Zone is real. Stalker's faith is thus justified. When the daughter uses her telekinetic powers we hear the dog make this noise (almost a high pitched sound) that I can't describe. The dog clearly doesn't like what the daughter is doing: that what symbolizes the doubt of Stalker in his faith, doesn't like the manifestation of Stalker's faith/belief as reality, as truth! This drives the point home for me. The dog symbolizes something negative that Writer and Professor (the non believers) instill in Stalker, the doubt in his faith/belief.
This ties back to the three of them standing in front of the Room, where we see a hopeless Stalker losing the physical battle against the two. Here we sense that he is starting to doubt himself.
I hope this wasn't too long of a read, I wanted to do this right. It's the least a masterpiece as Stalker deserves. If you can watch a film like Stalker I'm sure this too won't be too much for you.