Can I re-gift Steam gifts?

Yes you can re-gift steam games, as long as you accept them as gifts into you inventory, not activate them for yourself.

Games won't expire, but some items do (e.g. Coupons, beta access). Items that will expire say right in the description how long until they do. My coupon for 50% off Duke Nukem says

(Valid until Mon Jul 01 2013 02:59:00 GMT-0400 (Eastern Daylight Time))

After that, I believe it will disappear.

After changes by Valve, the accepted answer is no longer valid.

Steam gifts that are redeemed by their recipient are now automatically added to their Steam Library. There is no longer an option to redeem to inventory. The only exception to this rule is DLC for a game you do not own, in which case the DLC is added to your inventory, where it can optionally be traded or redeemed later.