Will downloading games slow down my laptop's internet?

I have a lenovo G510 laptop, 8GB DDR3 Ram, i5 4200m cpu 2.5ghz 3.1ghz in turbo mode, and an intel hd graphics 4600. I currently have Skyrim, Assassin's Creed 2, Assassin's Creed Brotherhood, and Assassins Creed Revelations installed. I'm afraid that having all these games installed will slow down my internet. Once I finish playing them I will close them and steam down. I currently have 850mb of 1tb remaining in my hard drive. Will having all of these games installed slow down my laptop? I don't think it should since they won't be taking up internet when they aren't running and there's plenty of hard drive space, but I'm just a little paranoid.

Yes. That little HDD space will definitely cause problems, as it fragments the HDD and eats into your memory paging space (usually 2GB or more).

I'd be shocked if you can play Skyrim at all.

Yes it will slow it down.

Very paranoid.
Unless a game is played (running) On-Line there's no use of the Internet.
Your internet speeds are determined by the IS Provider.
Programs that access the internet may be another problem.
Running in the background may be searches for E-mails, program updates, security searches/updates, and of course the possibility of a Virus/malware if you are infected.
Filling your hard drive will affect caching of pages.
Use an external drive (or flash drive) for documents and static data to save space.