Should the dependency be included in the repository?
I think dependencies should always be included in the repository as long as including them does not violate any terms of use. Few things are more annoying than having to find the right versions of the right dependencies manually before you can make a build. Sure, this is easy when you have automated tools to do this for you, which can find and download the right dependency, but what if you aren't connected to the web at the moment or the server is down or the project of the dependency has been discontinued completely and taken offline? Always include the dependencies if possible.
Should the dependency be built from within the same build script as
the rest of the project, or from a separate build script?
Unless there's a good reason to compile from source, use precompiled versions.
And why not provide options in the build script? A simple switch to choose whether the dependencies should also be compiled or not. If the user chooses to compile the dependencies as well, then just invoke their very own build scripts from your product's build script. So the user can invoke the dependcies' build scripts manually or choose to create a full build of everything. But I'd just deliver the dependencies as binaries if there's not a good reason to compile them from the sources. I think in the Open Source world, some licenses require you to distribute the sources along with your product but that does not mean you can't have them precompiled.
In short: Provide a whole standalone, working package if possible. This will provide the most convenience to your users.