I combine, compare, and contrast the results from two different numerical techniques (grid vs. particle methods) studying multi-scale processes in galaxy and structure formation. I produce a method for recreating identical initial conditions for one method from those of the other, and explore methodologies necessary for making these two methods as consistent as possible. With this, I first study the impact of streaming velocities of baryons with respect to dark matter, present at the epoch of reionization, on the ability for small halos to accrete gas at high redshift. With the inclusion of this stream velocity, I find the central density profile of halos is reduced, overall gas condensation is delayed, and infer a delay in the inevitable creation of stars.

I then combine the two numerical methods to study starburst outflows as they interact with satellite halos. This process leads to shocks catalyzing the formation of molecular coolants that lead to bursts in star formation, a process that is better captured in grid methods. The resultant clumps of stars are removed from their initial dark matter halo, resemble precursors to modern-day globular clusters, and their formation may be observable with upcoming telescopes.

Finally, I perform two simulation suites, comparing each numerical method's ability to model the impact of energetic feedback from accreting black holes at the core of giant clusters. With these comparisons I show that black hole feedback can maintain a hot diffuse medium while limiting the amount of gas that can condense into the interstellar medium, reducing the central star formation by up to an order of magnitude.