this is fun, great job! there is a definite fun factor to jumping through the bullets, and the boss at the end was cool.
for sure, there needs to be more feedback when the player gets hurt. it's one of the most important things that can happen in the game world, and the player needs that feedback to get a feel for how well they're doing without having to keep glancing at their health bar.
a checkpoint before the boss would have maybe helped it feel more fair, but you don't necessarily need it, depending on what type of game you are trying to make. when i died to the boss and got sent back to the start, the reason it felt unfair to me was because i had to do the same thing over again, and there wasn't really room for me to do it better or faster. if it was rewarding to improve my skill at the game, then it would probably be fun to play again. so, for example, instead of giving the player 5 health for each level, it might instead make sense to give them 5*4=20 health for all four levels (or however many there are), but the health doesn't regenerate when they enter a new level. that way, if i can just perform better on my second run, i'm rewarded for my skill progression and i have a better chance to beat the boss, because i'm saving health throughout the different levels in preparation for the boss.
just thought i'd give some feedback! i had fun playing it. thanks for making it!