If there is space, I would go with a full desktop (or desk-side server case) any time. I totally appreciate the argument for a laptop, and if I were to have just one machine then it would probably be a laptop but, in a workshop a full-size machine has too many benefits. My workshop machine is an old Compaq Deskpro compact, with "proper" legacy serial and parallel ports. It is fine for burning flash via JTAG etc, PDFs, printing and so on. It is also very very quiet and great for hacking around with hardware. Generalising, it is quieter than many laptops, which tend to have heat problems and so run the fan quite hard.
Talking of which, heat is the problem when rending videos on a laptop. You can run a quad core i5 at full speed all day in a desktop, a laptop will (in addition to costing you a lot more) tend to throttle back as the heat soak builds up. They are also much better at running 24/7 if you want to (for example) run a web cam for security, and not being moved around tends to make them more reliable. You know the arguments in favour of maintainability and upgradability with desktop PCs. If need be, it could live in that electrical utilities cupboard, securely out of sight.
I guess the limiting factor is broadband speed, but you could remotely edit (and then render and upload) video on your lab PC, from home via remote desktop. I make the dangerous assumption that the lab broadband will have a better connection then at home. Either way, it is still nice to be able to access your lab software remotely.
Pencil and paper may not offer mathematical simulation, but they are a heck of a lot better for "rapid prototyping". Ultra high resolution, erasable, no power usage, carbon neutral, and the ultimate intuitive interface!