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Whichever distro you're most comfortable with.
A Linux distribution, known as distro or flavor, is an operating system that uses the Linux Kernel. I think the most common one is Ubuntu.
It depends on the Linux you are running. There are thousands of Linux distros. It is possible that your distro wasn't intended to run DVD-ROM. But most Linux will run them.
Depends on the distro, but most use wither bash or sh.
Most modern Linux distributions have a method of updating the system without requiring a clean install. The exact procedure for this varies by distro.
Almost every Linux distro can run multimedia programs, and has them available in their repository. Several distros are targeted at multimedia production and bundle such software by default. These include Ubuntu Studio, 64 Studio, dyne:bolic, and Musix.
There are many different websites that distribute a variety Linux operating systems. The most popular place for desktop distributions computers is Ubuntu.
There is a link to the download page in the "Related Links" box below. Note that the site, and thus likely most of the distro, are Chinese.
The Linux kernel supports up to the latest Intel and AMD processors, as well as some other architectures. As such, most programs should run about the same on all of them.
There's Ubuntu, Debian, Red Had, Fedora, Gentoo, Arch, Mint, SuSE, Slackware... A good way to find out is to look into Distro Watch. The web site more or less keeps a monitor on how popular a given Linux distribution is.
If the hardware is physically enabled (such as via a switch or in the BIOS), most Linux distributions should be able to use the Bluetooth receiver. The program that is used to manage Bluetooth connections varies by interface and distro.
Creating a Linux distro is a fairly involved task, which is why it is usually done by teams and not individuals. Assuming you were to create your own from scratch, you would need to compile a Linux kernel and related programs, and write your own init scripts. Try compiling just the Linux kernel, and you'll see why few people bother to build the entire thing themselves. It would be far too difficult to remain current with most software packages. If you want to distribute it to other people, you'll also need to figure out things like warranty, support, and update management. LFS (Linux from Scratch) is an excellent guide on building a very basic Linux distro yourself, and learning what is involved in the process. For a simpler process that does not require too much time or programming knowledge, look into creating a distro based on an existing one. Many distros offer automated or semi-automated tools to help you do so.