Free games with no download

Yes. Browser games run inside the web page itself, so there is nothing to install on your phone or computer. Plixoo is built this way throughout - every game on the site plays without a download.

Browser games are generally safer than installed apps, because they run inside the browser's sandbox and cannot access your files, contacts or other apps. Plixoo also requires no account, so there is nothing to sign up with.

They usually do, because nothing needs installing and no administrator permission is required. Whether a specific network allows the site is up to that network's filtering.

Almost none. A game loads into memory like any web page, and the only thing kept afterwards is a few bytes recording your best score.