Gigabit with cat5 cable

You can use Cat5 for really short runs in a Gigabit environment, but it isn't recommended. Cat5e is fine in almost all scenarios, but if you're wiring from scratch, use Cat6, it's only fractionally more expensive.

Good Wikipedia info here - particularly the link about "far-end crosstalk".

Theoretically, CAT5 should suffice.

Practically, most switchs will require CAT5e or higher (CAT6 etc.) to work properly and will throttle back to 100m if you try to hook them up with CAT5 or low quality CAT5e cabling.

Higher spec cables are not expensive so most of the times I'd recommend the upgrade.

1000 Base T was designed for use on standard CAT5 cable, however CAT5E is recommended due to its tighter specifications. I'd have a read of the Wikipedia article on CAT5.

A Gigabit link might "come up" on a CAT5 cable, however you may notice increased retransmits on the interface, due to interference issues.

Definately possible, but not recommended.