Electronic – Kicad PCBnew: Cannot figure out why DRC reports unconnected pads

I didn't know it could do this, but I've just tried it out, and it works. Here's how I did it.

1. Open a new schematic, or whichever schematic you want to add an existing design to.
2. File->Import
3. Select the schematic of the design you want to add to the open schematic.
4. Make sure the nets are named the way you want (identically named nets will merge)

That's it, I get a new sheet for the added schematic and the completed layout added next to the existing layout.

If this doesn't work for you, it's likely because you're using the Eagle Freeware, not Eagle Standard or Professional. The hobbyist/non-commercial version is not too much if that is the problem and it counts as Eagle Standard. It's what I have.

You can split your design into a hierarchy (e.g. multiple pages), then have local nets to each page and also global nets. You can also use specific page to page connectors (so the page symbol is like a component you can drop into another page (over on the right hand icon set you will see an icon labelled "Place hierarchical pin in sheet" and another named "Place a hierarchical label" and a couple of other related icons)
Have a play around with the demo projects to see how it works. There are also some good tutorials out there that go through this stuff. Tutorial 1, Wiki Kicad, Hierarchy Tutorial.

Go into wherever Kicad is installed (e.g. Program Files/Kicad/Share/Demos), and look in the Demos folder for plenty of examples of different ways of doing things.