A few days ago I accidentally (yes, it actually was accidental) downloaded a strange file. I almost immediately found the file on my desktop and deleted it. Then I ran a full sweep with Spysweeper, and it turned up the normal stuff.
The next day, my dad wanted to change a security password of some sort on my D-Link adapter. Before he did, he attempted to access Internet Explorer (even though we all use firefox). As we were locked out until he put the new password into my computer, it didn't show the page. But the URL, I saw, was for some kind of virus protection download. There was even a message telling us that we were infected, but this may or may not have all been phony stuff. After the new password was in, I got into Firefox without a problem.
Today, I found some weird word files in my documents (they looked faded and their titles were the same as some that I had made, but these also had symbols that I don't think you can put in titles) and risked opening one. First, it a small window told me this "type" of file could not be opened without a converter that needed to be downloaded. I clicked the little no thanks button, and I saw the "unconverted" file: some weird word and a bunch of squares. Then, Firefox opened up, saying I was infected and wanting me to download some stuff. But then there was a phishing alert in the corner. This was the first time I had ever seen this, even though I had been to sites like the one I had just seen before. I closed it and deleted the strange word files.

So here are my questions:
Does it look like a virus of some sort,
And if I do
Is there any way to get rid of it, or do I have to reformat the entire computer
Can it spread through network magic to the other computers connected to it
Why didn't spysweeper pick it up?

Any and all help is greatly appreciated!

The Word files may have been ones Word creates as temporary files while you're working on a document that didn't get cleaned up. The faded part sounds like the way hidden files are rendered when you've enabled viewing of hidden files.

The popups telling you to download things sounds like one of the typical symptoms of a spyware/adware infection. You may need to scan with a full antivirus product and multiple spyware products for it to be detected. No single product is 100% effective. Spreading may be possible depending on what it is. Just unhook the computer's network cable until you've confirmed the computer is clean or unhook the other computers if you need to download something. A reformat isn't necessary to remove an infection, however depending on the particular infection, it may be less time consuming and certainly effective.

Manual removal can be a challenge. Why don't you start first with automated tools before attempting a reformat. These boards are filled with posts from people who never get their computer quite right again after attempting a complete rebuild. These sticky posts discuss products for elimination/detection and techniques for preventing infection:
Must Have/Useful Programs
After the Storm (The virus is gone)