How to put steam and games in an EHDD?

I'm not sure how that works. Most games don't, I know, because installing a game does write something in the registry and so on. You would need to have your steam install on the HDD first to get the steamapps folder there, or so.
Half-knowledge, sorry.

Most games couldn't care less or will simply write what they need to each machine they're played on. A few are pickier and have to be installed properly, but over the years we've copied and moved games between locations on the same machine, other machines on the LAN, or to new hdds and rarely has a game noticed or cared.

Doom 3 fussed a bit, iirc, as did UT2k4, but nothing else sticks out in my mind, and a few games would ONLY work on certain pcs after being copied, as the installer was locked by OS but the game was fine.

IOW, try it, and you might be pleasantly surprised.