Say for instance, you have your own personal library of your own code you like to re-use. If one uses their own code in their own personal library, do they have to then bundle that library with the application when they are finished in order for the application to work? or do people just do a cut and paste job from their library into a seperate library for said application?
Is it more common for professional developers to have one giant library they use very often? Or lots of tiny specialized libraries?
You'll need to bundle the assembly in your application. Never, ever, ever, ever cut/paste your code from one project to another. Keep your library in its own solution. If you must, either add the project into the solution you're working on or compile your library and reference the assembly. The first way is more efficient because it generates the assembly for you. However, if you don't want someone else touching your library and making mods to it (open/closed principle), bundle the assembly rather than the project. You'll need to have the assembly (DLL) regardless.
Is it more common to have one big library rather than a tiny one? That answer depends wholly on what you're coding. I have one library I use for everything: it carries my utility extensions, I use it for Active Directory searches and a bunch of other things. It's in one assembly. I could have bundled it in multiple assemblies but I hate managing multiple dependencies in the same project. However, I develop mostly WinForms and ASP.NET, where I have more room to play with in terms of computing resources. If I were go to mobile, I'd start partitioning the assembly into smaller ones so as to use only the ones that I need. Less assemblies, small application size thus leaner application.