Will minimizing or hiding my browser window affect my download speed?
Answer: The simple answer is no. Whether or not you can see a web browser window should have no effect on the download performance. Just because something is hidden on your screen doesn't mean the CPU isn't working as hard in the background. Downloads happen in the background anyway, so the speed should not be affected by the visibility of the application.
The reason I say the speed "should not be affected" is because there are some applications that do in fact go into a reduced performance mode when they are not active. OS X Mavericks and later includes a feature called "App Nap," which reduces the resource consumption of inactive applications. From my experience, hiding my browser window still does not affect download speeds on my Mac. I've tried minimizing browsers on Windows machines and it has not seemed to affect the download speed. Still, it's not impossible that either the operating system or application could throttle data transfer rates when the application window is not active.
To be sure that minimizing your browser windows does not affect your download speeds, you can use your operating system's built-in monitoring tools to check the speed. In Windows, you can open Task Manager (Control+Alt+Delete), select the Performance tab, then click the "Resource Monitor" link near the bottom of the window. Click Network to see a live graph of the download speed.
In OS X, you can open Activity Monitor (Applications/Utilities/Activity Monitor), the click the Network tab at the top of the window. The bottom of the window will display the network traffic levels in realtime.
Minimizing or hiding your browser window shouldn't affect your download speed, but using the monitoring tools will let you know for sure.