Can I go to a font site (1001freefonts.com), go through all the fonts, download them all (come in zip format) and then group them together into one big .zip file and redistribute on my site? I would keep all of the files that are inside the .zips intact.
No, you can not. At least some of the font owners prevent you from doing that. They allow you to list the fonts for free, but it is a restricted free usage. For example Ray Larabie explicitly states that on his web site. He does not allow you to pack it into one zip file and allow the easy packed download.
This is the wrong place to seek askwers to that sort of questions. It's the 1001freefonts.com people to give you instructions on how you may use the content you collect from their website.