We study the lattice structure and dynamics of the quantized vortices in superfluid helium-4 using a new rotating experiment. This setup includes control of the entire apparatus from the rotating frame, installation of a new EMCCD camera that allows for imaging of nanoscale tracer particles, and the development and implementation of a new isolation cell, which permits investigation into new phenomena such as differential rotation in helium-II. We have observed the vortex lattice dynamics in the (r, &phi) plane (i.e. transverse to the vortices) and present here the first real-time visualization of Tkachenko waves in helium-II from this cross section. Additionally, we present evidence of differential rotation with distinct Stewartson layer boundaries, possible Kelvin-Helmholtz instabilities, and the formation and propagation of superfluid collective vortex eddies. We show that the angular velocity is a function of radius and may be driven by the geometry of the isolation cell. We also document the observation and analysis of gravity-capillary surface waves that demonstrate an interaction between the liquid helium free surface and the bulk of the fluid.