Undecidability of the PCP problem with bounded width
If $w$ is bounded, and the alphabet size is bounded, then the number of possible words is bounded. Thus, there is only a finite number of possible instances, so the problem becomes a finite languages, and hence decidable.

If you don't bound the alphabet, then you can encode any width of $w$ by adding more letters, so the problem is the same as standard PCP, and hence undecidable.