Yes, both audio and data CDs can be duplicated. You can even create audio CDs that are compilations of other audio CDs (perhaps a personal "best of" disc).Bear in mind that most CDs are protected by copyright laws.

Yes and no. The process can be a bit more involved than that, and requires software that (usually) comes bundled with the drive.
With "packet writing" software, and a recorder that supports it, you can treat a CD-R or CD-RW disc like a floppy. On a CD-R you can only write to each part of the disc once, so deleting files doesn't free up any space. There are other limitations as well.
With more traditional software -- necessary if you want broad compatibility -- you usually end up writing everything to the disc all at once. When you're doing the writing you can't interrupt the drive, and you can't reclaim the space you've used. If you want to write your files in smaller bunches, you lose a fair bit of space every time you stop and start again.

Not directly. CD and DVD are very different formats, so you can't write DVDs with your CD recorder. You may be able to convert the contents into a lower-quality format though. Be wary of scams. See section (3-49).
There are devices now that can record both DVD-R and CD-R. Those are usually advertised as "DVD recorders", not "CD recorders".