This is a good start, but it has some balancing issues. In this style of Shoot 'em up, where enemies are tough, come in swarms, and move forward mostly in a straight line, and you shoot only in front of you, destroying foes punctually before they overwhelm you by filling the entire screen is a top priority. This is a trait shared by many shmups of this style, I've noticed. Neglecting even one enemy would spell trouble, since more enemies would appear behind it, effectively blocking off that column of space, since you can't attack sideways, and you don't have enough time to jump in and destroy a ship before it crosses the lower boundary. This limits your movement space, and eventually you'll be confined to a single narrow column which is not populated by enemies.
In your game, given the toughness of the enemies and your relatively low damage output,
this trap is very difficult to prevent.
I don't know if you have different plans in the future, but right now, it's suggested that the goal of the game is to destroy enemies, since you're awarded points solely for defeating enemies and not for simply surviving and dodging. This is a problem, though, because the enemies are rather hard to kill, and since you don't have much HP, you'll be spending most of your time dodging rather than killing. In my playthroughs, I found myself killing only the weak blue enemies while dodging all the rest, then hoping for a box containing a good upgrade.
One thing that you have to keep in mind is that an enemy's SPEED, not just its HP, factors into its endurance. The faster an enemy is, the less time you have to move in and the fewer shots you can get in. This is especially noticeable with the shooting enemies, since they are rather fast and still take four hits to kill. It's even worse if you want to take time to dodge its bullets. Same deal with the orange enemies that move back and forth: its movement pattern already causes most of your bullets to miss, so it should probably get less HP. A good way to measure this "endurance" is to see
how much of the screen you need to destroy it. This means that if you start shooting at it when it first appears at the top of the screen, and don't destroy it until it's more than halfway across the vertical length of the screen, maybe that's a bit too much HP.
So basically, the objective of the game is to kill enemies, so you need to either build around this objective, or change your objective.

I suggest using one or more of the following:
- decrease enemy health
- increase your rate of fire
- different firing patterns than directly in front of you, such as spread shot (this might be a powerup already, but I haven't encountered it)
- more varied enemy movement, e.g. diagonally
- higher missile capacity, and/or missile regenerate naturally over time (or some way to acquire missiles other than killing foes)
- more abundant powerups, and are temporary (to balance it out)
- award points for amount of time survived
Bug Report:
- after the bomber passes, the sound of bombs being dropped keeps going (my guess is the bomber is not reset)
- after a while (about three minutes in), enemies stopped appearing, besides the purple ones and powerup boxes, then after a bit those stop coming as well (could this be due to enemies being spawned with 0 or negative speed?)
- not a bug, but double bullets makes the game lag dramatically (maybe something can be done?)
- one of the powerups makes it so your bullets can't do damage
Also, a description of what each item does would be useful; I can't figure out some of them, such as the curve thing dropped by the shooting enemies.