Yes, but they are extraordinarily expensive.
The Velocity Micro Slot Loading DVD/CD player sold by Costco is an external drive that fits this description. Most slot loading dvd/cd drives are made for high end laptops however they are available for any application.
In my experience, the laptops have slower drives in terms of read speed.
The majority of Laptops on the market as of June 2012 contain a removable DVD drive as standard. For a few years it has been common for even the cheap budget laptops to contain an optical drive that could read DVDs but could only write to a CD, but now even the most budget laptops contain a DVD writer as standard. The Smaller Net-book computers and some of the New Mac Book laptop models seem to be some of the few exceptions to this by containing no optical drive at all. Although DVD drives are now being replaced in most of the top end models of Laptop with a Blue-Ray dive, as these are backwards compatible with DVD drives it could be argued that they also still do contain a removable DVD drive.
Any that are USB.
There are drives that are somewhat quieter than others, but this is more a function of speed. The slower the drives spins, the quiter it is. LG, ASUS, and Lightscribe have some noted near silent drives.
Yes, the Panasonic DVD-RAM Cartridge Drive uses DVD's and is an excellent unit.
As many as needed. In laptops/notebooks there is usually only one hard disk. In a desktop or server there could be potentially any number of disk drives. If you mean a CD/DVD drive the same applies.
The lens in the DVD player may be dirty and unable to read discs. And some laptops cannot play DVD-r's.
If it has a USB port, you can use memory sticks/flash drives. I don't know of any laptops that don't have floppy drives, zip drives, CD ROM drives or USB ports.
Well, that depends: Do you have any handhelds, TVs, DVD players, ipods, laptops? count them.
Optical drive is just another fancy name for DVD drive or CD drive and it is usually found in in laptops and computer and is used to play any tyoe of DVD or CD.
There are plenty of companies that make DVD drives specifically for burning DVDs to the computer. Plextor is known for this.