abstract: High-energy explosive phenomena, Gamma-Ray Bursts (GRBs) and Supernovae (SNe), provide unique laboratories to study extreme physics and potentially open up the new discovery window of Gravitational-wave astronomy. Uncovering the intrinsic variability of GRBs constrains the size of the GRB emission region, and ejecta velocity, in turn provides hints on the nature of GRBs and their progenitors. We develop a novel method which ties together wavelet and structure-function analyses to measure, for the first time, the actual minimum variability timescale, Delta t_min, of GRB light curves. Implementing our technique to the largest sample of GRBs collected by Swift and Fermi instruments reveals that only less than 10% of GRBs exhibit evidence for variability on timescales below 2 ms. Investigation on various energy bands of the Gamma-ray Burst Monitor (GBM) onboard Fermi shows that the tightest constraints on progenitor radii derive from timescales obtained from the hardest energy channel of light curves (299--1000 keV). Our derivations for the minimum Lorentz factor, Gamma_min, and the minimum emission radius, R = 2c Gamma_min^2 Delta t_min / (1+z), find Gamma < 400 which imply typical emission radii R ~ 1 X 10^14 cm for long-duration GRBs and R ~ 3 X 10^13 cm for short-duration GRBs (sGRBs). I present the Reionization and Transients InfraRed (RATIR) followup of LIGO/Virgo Gravitational-wave events especially for the G194575 trigger. I show that expanding our pipeline to search for either optical riZ or near-infrared YJH detections (3 or more bands) should result in a false-alarm-rate ~1% (one candidate in the vast 100 deg^2 LIGO error region) and an efficiency ~90%. I also present the results of a 5-year comprehensive SN search by the Palomar Transient Factory aimed to measure the SN rates in the local Luminous Infrared Galaxies. We find that the SN rate of the sample, 0.05 +/- 0.02 1/yr (per galaxy), is consistent with that expected from the theoretical prediction, 0.060 +/- 0.002 1/yr (per galaxy). / Dissertation/Thesis / Doctoral Dissertation Astrophysics 2017