The bullet hell things really depends on the game engine. The engine used for this game is custom-made and lightweight, so it won't do a lot of other transform logics creating unnecessary overheads. Actually you can do this with glfw or other similar graphics libraries.
The bullet logic is calculated in some way that the engine don't have to frequently update the bullet states. Every bullet has bullet state consists of position, velocity(in axial speeds or rotation+speed) and acceleration values (angular velocity+linear acceleration), and the position and velocity values are both updated every frame.
As for spawners, there are no actual spawners, but all bullets have an option of being spawned from any of the enemies. As I use Lua for the game script, every "spawner" is actually one cycle of a for loop, with a series of tasks and bullets assigned to a spawned enemy.
The tempo change was designed as part of the introduction, as I thought it's easier to start slower and gradually gains speed; from your feedback I feel this actually bad for new players. If there are more stages, I would definately give a constant tempo for all early stages.