Is there any visual indicator that the pirate will be attacking you?

If the pirates are already after someone, there will be a scimitar by their name on the pirate screen.

Does outbidding the current bidder entice the pirate to attack someone else?

Yes.

What does the horizontal bar indicate under the bid amount?

That sounds like you're referring to the 15 minute pirate countdown. At the end of every 15 minutes (exactly how many remain is to the left of the bar) there's a bidding war, and pirates attack whomever has the highest bounty at the exact moment they launch the raid, which will be 30-90 seconds from when the countdown ends.

If you have Diplomacy/Trinity you can research options to send pirates on raids to particular planets in addition to their normal raids.