Here’s my take – as Mayhew said, no you don’t have to replace the chain. A little rust isn’t a big deal.
As for what lube to use in cold, wet, salty conditions, maybe you could get by with frequent relubing using a wax-based product. I usually use waxy stuff in the summer, but for commuting in our current lovely conditions (minus yesterday), I stick to a thicker “wet-style” lube.
Keep in mind that doesn’t necessarily mean your chain will be cleaner – the wet-style lube picks up dirt more readily than a teflon-based dry-style or wax. But for below-freezing temperatures, I have found that the wet-style lube keeps more of its lubricating properties. Just make sure to wipe the chain well after applying.
Eh, whatever, though – I’m not an evangelist on this stuff. This is just what I’ve gotten used to doing.
The late great Sheldon Brown has some essential chain-cleaning advice here.