1) I guess we're to use SSH2 and not SSH1,
SSH1 and SSH2 are versions of the protocol, in particular the initial handshake. The SSH1 had a vulnerability. Which protocol is used is determined in the initial handshake, for compatibility. A prodent server administrator disables SSH1 for sshd.
but does it matter whether we use DSA or RSA? Is one much slower than the other?
I don't think it matters much. I vaguely remember DSA was introduced because of the RSA, Inc. patent.
3) To use Putty on my work computer (besides my one at home--where I've just created the key), should I create another private/public key pair, and append the public key to my authorized_keys/keys2 file, or just transfer my private key to the second computer (via floppy disk, etc.)--what is the more usual/accepted practice?
It's probably a matter of taste. I personally would use two keys. And I wouldn't use a key on my work machine unless I'm totally and utterly sure it can't be compromised.