Weird. That shouldn't be possible. Maybe you haven't noticed that there is armor bar below health bar. Armor acts as health and when you take damage, you lose armor first. Shield thing gives you 1 point of armor. Let me know if that helped.
Can confirm the invincibility issue. Not sure if it was caused by a shield, but I noticed my invincibility while fighting the boss on 1-10 and it persisted onwards past then. Not sure when exactly the bug started on the run. Bullets still played their particle effect and disappeared after hitting the character, but no damage was dealt.