If I receive Bonus Chips for my first registration and get eliminated before the Early Bird Bonus Chip window closes - will I receive the bonus again if I re-enter?

Yes, however, the % will depend on which blind levels you re-enter at and if re-entry is available in that tournament.

If I receive Bonus Chips for my first registration and get eliminated after the Early Bird Bonus Chip window - will I receive the bonus again if I re-enter?

No, as this is a new entry and after the cut-off window.