They are not dead and still part of the Framework.
If you want to be able to create modular like components that you can move around into different zones on a page then you may still want to use WebParts.
Yes, but doing things the web-parts way means surrendering a lot of control over the resultant markup. They were released in 2002 before web standards became popular, and they didn't receive many (if any at all) changes in the 2.0 release of ASP.NET.
And if you're developing something that should be accessible you probably should avoid them (apparently certain public websites have to meet web content accessibility guidelines or you could be breaking disability discrimination laws).