How early can I apply for postdocs? I am about 1 year away from finishing my PhD (hopefully), and have recently seen a few advertised postdoc positions that I think would be perfect for me. Most positions indicate that the start date is around 3 months away. My question is: how early is too early to apply? Is it worth applying for these positions and indicating in my cover letter that I wouldn't be able to start for another year? Is it OK to email professors directly and inquire informally about open positions in their labs at this stage?
You certainly can apply early - the worst they can say is "No", which is functionally equivalent to what happens if you don't apply. It's also a perfectly fine time to start asking around.

The answers will depend very much on the postdoc. For example, right now I have a postdoc position in my lab where "A year from now" is 1/3rd of the way through funding, so it likely wouldn't be acceptable, as I needed a postdoc working right away. On the other hand, there are other positions where starting would be considerably more flexible.