Rebuilding fence - lumber questions
Wood isn't pressure-treated just because it may touch the ground. It's a sponge, and it'll always be damp inside unless you're in a desert. Whether you need it is your prerogative, but why would you pay a premium for redwood pickets and not use something of roughly equal longevity to support them?

Brand recommendations are off-topic here. I've removed that portion of your post.

Screws don't need to be protected. They're presumably corrosion-resistant, or they're not suitable for a fence. 

Important: I don't know what post bases you're planning to use, but most aren't designed to hold up a fence. They're merely connectors and/or spacers. Be sure yours are up to the task.