Quad trees are a totally different thing than the collision method. Quad trees are a form of spatial partitioning. Spatial partitioning helps weed out distant objects when trying to find pairs of colliding objects or testing ray collisions. Spatial partitioning is technically entirely optional, but in practice is required for performance.
Continuous vs discrete determines how often collisions are checked for. Continuous collision detection works without stepping time, instead checking to see when the next collision would occur and telling you what time that collision will be. Some folks also call swept volumes a form of continuous collision detection.
The primary advantage of continuous collision testing is to avoid tunneling. Tunneling occurs in discrete systems because objects are only checked for collision at specific intervals. If an object is small and fast enough, it may be completely on one side of a wall at one time and the next increment it will be completely on the other side. Continuous collision detection also provides more accurate results, as it can tell you the precise moment two objects will collide, while discrete systems only find out about collisions after the objects have penetrated each other.
The primary advantage of discrete detection is that it tends to be much faster and simpler for objects with complex shape or large size. Continuous collision detection requires mathematical descriptions of all objects and their motions and the solving systems of equations, which can be difficult, slow, or outright impossible in many circumstances.
Swept volumes are a way of using discrete detection while solving tunneling. You can take an objects current position and old position and enclose a shape around both of them ( this would be a line if your objects are points, a capsule if your objects are spheres, etc ). It is not as accurate as continuous collision, but simpler.
In general for simpler physics engines, use discrete methods for most objects and use a ray cast (continuous) for bullets. Specific games might need to use continuous or swept volumes more often, depends entirely on the specific game.
For things like your machine gun, consider simply reducing the number of bullets. Players really aren't going to notice 100 individual bullets when you can "cheat" and simply draw a bullet spray wffect and use a much smaller number of rays for actual hit tests.