Can the old 20gb hard drive be attached to the new console?

Nope. Hard drives from the old fat console don't work in the slim and vice versa.

In order to have more memory for downloads then, would it be possible just by hooking a flash memory stick into the USB port?

The 4GB Console has a Flash Memory inside of it, no actual hard drive is present. That being said, you could use USB flash drives to store games on it, however you should be aware that certain games require an actual hard drive to be present to play them. Two such games that require this are Halo and Black Ops due to their Theater features.

If you look around online, Amazon for one, you can find a 250GB hdd for the slim console for less than $100USD. Just be sure it is an official MS hdd, it will have their logo on it and the packaging.

You can use a USB flash drive, but it's only limited to 16GB of memory. You can buy the official Xbox 360 USB flash drive (http://is.gd/3kLtLi).My recommendation is you to buy the 250GB HDD. You can buy it on Amazon.com (http://is.gd/dmAkvT)

Would I be unable then to play black ops live with the 4gb console?  Even if I have a flash drive attached all the time with the maps or whatever?

The flash drive will work pretty much exactly the same as a hard drive would. You can play Black Ops online and download any additional maps to the flash drive just as if a real hard drive was present.  There are a small number of titles that do require a real hard drive though, but Black Ops is not one of them.

Unfortunately, you can't use an Xbox 360 Hard Drive with an Xbox 360 Slim. Sorry :(