is it possible to create parent-child relationships between sub-sites in SP2013? For instance if I set up a master Site called "Projects", and then set up a sub-site in "Projects" called "Website design for X". Could I then create a sub-site that would be a child of "Website Design for X", something like "Design wireframes", so that when I am in "Website design for X" I can see all items that are also part of the sub-site "Design wireframes" ?
Yes, you can create a site hierarchy like the one you describe: 1) Projects --> 2) Website Design X --> 3) Design Wireframes
Content Rollup from the sub-sites can be done using Search or Content Query.
However, you should first rethink if it is really necessary to build a site hierarchy of three levels. Creating "Design Wireframes" as a separate Document Library or Picture Library within "Website Design X" may be sufficient in many cases. This would reduce complexity for users while browsing. At the same time your content rollup could be done much easier.