HEAD DOWN, HEAD FIRST: At the end of your turn, choose a side of the square Bulldozer occupies. Lines of fire drawn to Bulldozer that pass through that side of the square are blocked until your next turn.

Would a character using Sharpshooter be able to ignore "Head Down, Head First"?

No. The Line of Fire would still be blocked. Sharpshooter only allows you to ignore opposing characters for Line of fire purposes. It would still be blocked by other game effects.

What if Bulldozer is not the intended target? Since the LOF is not drawn to Bulldozer but to a character behind him, wouldn't Sharpshooter still ignore Bulldozer?

HEAD DOWN, HEAD FIRST: At the end of your turn, choose a side of the square Bulldozer occupies. Lines of fire drawn to Bulldozer that pass through that side of the square are blocked until your next turn.

Yes it would.