Should the dependency be included in the repository?
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.