Thank you for the comment. I know the enemies could look better, I am still learning so I drew them quickly to get them working in the game. I wanted to go back and have another go but there wasn't time. To answer your question, it is NOT possible to win without dying. There are certain jumps that require the Yellow Boots.
Since you took the time to share your suggestion, I'll tell you what my original plan was. Currently, the player becomes more powerful after each death. I wanted that to be true of the enemies too. Eg if you kill a red scrambler by jumping on it, next playthrough it will be able to jump; if you shoot a yellow jumper, next time it will drop bombs as it flies; and so on. So the player gets more powerful after each death but also can spread those powers to enemies killed. This would increase variety and re-playability. That's a long way of excusing the lack of checkpoints :D