Nope. Not possible without a bit of hacking the hardware.
You’ll need to transfer your files, and probably pick up the version with more HD space. Welcome to gouging you one megabyte at a time, courtesy of Microsoft.
From what I’m seeing online and have heard from friends, yes. The big issue seems to be that Arcade games downloaded to the HDD are linked to the old machine’s serial number. You will have to be connected/logged in to Xbox Live to play them as is on the new console (“roaming”), or follow the directions here to transfer your licenses to the new console (which will require redownloading content, apparently).
ETA: first two responses are contradictory! Yay. I admit speaking from second-hand sources, so…
Sure. I have an old 20 gb drive that came in my old xbox, and when the old one crapped out, and I bought a new one, I took the drive out of the casing, but kept the adapter (they’re basically SATA drives) to hook it up to the Xbox, and just plugged it in where the new drives would go.
It’s unsupported (in just about every sense, hanging from the connector, and not being a sanctioned way to do things), but it seems to work just fine, and always has.