In the example you highlighted and indeed in most plane-wave DFT codes, there is periodicity in all three dimensions including for surface slab calculations. In the case of a surface slab, vacuum space is commonly added in the $z$ dimension. The vacuum space is there so that an adsorbate can bind of course, but it's also there because of the boundary conditions. A vacuum space ensures that the adsorbate-slab complex does not interact with itself over the periodic boundary, provided the vacuum space is large enough. In this way, you're modeling what is effectively a 2D system while still having 3D periodic boundary conditions as part of the DFT calculation. If an interactive example would be helpful, I recommend John Kitchin's DFT ebook, specifically Section 5.
The answer to both your questions is actually, for the most part, one in the same. The reason plane-wave basis sets are so useful is how well they lend themselves to periodic DFT calculations. In this case, you can represent a crystalline or bulk system with much fewer atoms than you would be able to do if you relied on a Gaussian basis set. A given metal might have a primitive unit cell of only a few atoms, whereas modeling the same metal with the same degree of accuracy without finite boundary conditions might require several hundred atoms to prevent edge effects and ensure the system is large enough. A localized, Gaussian basis set is not inherently better than a plane-wave basis set. The latter, however, is naturally better suited for adsorption problems in general.
As a side-note, because it can be helpful for to visualize orbitals in a Gaussian basis set, there are several algorithms that project plane-wave bands to Gaussian type orbitals, such as the periodic NBO code of Dunnington and Schmidt. This is mostly for gaining insight into the electronic structure of the chemical process, rather than a question of accuracy.