If I shutdown my laptop while an application is installing, will the application stop downloading?

yes sometimes it will pause but your best off leaving it on all night and if your worried about getting spy wear or a virus because your internet is on you can turn it off as long as your not downloading from the internet.

When you place your computer into "sleep" mode, it is essentially shut down. It just stores information in your RAM until you bring the computer out of "sleep" mode. Shutting your computer down while an application is being downloaded is not a problem. It's when the application is being installed and you shut the computer down, that can corrupt the installation and you'd have to start completely over. This can mess up what has been installed in the Windows Registry and cause fragments that will cause problems. It's best to let any software install once that operation has begun.