1. No it is not an app, it is an operating system. BUT, running a Linux virtual machine on your desktop using VirtualBox is possible. VirtualBox is the app, not Linux. It used to be possible in Microsoft Virtual Machine, but that is getting more and more difficult and not for the lay person.
2. Browsing anonymity is a function of your web browser. You can run the same web browsers in Linux you can in Windows except for the Microsoft browsers. It comes with Firefox installed. How anonymous it is depends on the browser settings, but out of the box it is anonymous.
3. The "Live CD" (DVD actually) that you install Linux from boots into a full Linux without changing anything on your computer. You can test drive it from the Live CD. There is an "Install Linux" icon on the desktop. You can install Linux on any removable media including a thumb drive or external hard drive, but it may not be the best user experience. You can also install Linux as a dual boot OS, meaning you choose which OS to run when you turn on the computer. When you dual boot into Linux, your get the full power of the computer at your disposal. Performance is somewhat limited when you boot off a thumb drive or external hard drive. Although, some of the new thumb drives are pretty fast.
4. Which search engine you use is a browser setting, just like in Windows. I use DuckDuckGo which does not track your searches. Out of the box Linux Mint uses the Yahoo! search engine with Firefox. I use DuckDuckGo with Chrome. There are two versions of Chrome, one is Chromium which is an open source browser and more Linuxy. And you can use Chrome which is exactly like the one on Windows.
5. Encryption is an option during the installation. You can use a fully encrypted file system if you choose. On the web, it depends on whether the site you go to uses http (unencrypted) or https (encrypted), and that is independent of what OS you use.