Black hole (BH) accretion flows and jets are qualitatively affected by the presence of ordered magnetic fields. We study fully three-dimensional global general relativistic magnetohydrodynamic (MHD) simulations of radially extended and thick (height H to cylindrical radius R ratio of |H/R|~ 0.2--1) accretion flows around BHs with various dimensionless spins (a/M, with BH mass M) and with initially toroidally-dominated (o-directed) and poloidally-dominated (R-z directed) magnetic fields. Firstly, for toroidal field models and BHs with high enough |a/M|, coherent large-scale (i.e. >> H) dipolar poloidal magnetic flux patches emerge, thread the BH, and generate transient relativistic jets. Secondly, for poloidal field models, poloidal magnetic flux readily accretes through the disk from large radii and builds-up to a natural saturation point near the BH. While models with ||H/R|~1 and |a/M| 0.5 do not launch jets due to quenching by mass infall, for sufficiently high |a/M| or low |H/R| the polar magnetic field compresses the inflow into geometrically thin highly non-axisymmetric ``magnetically choked accretion flow'' (MCAF) within which the standard linear magneto-rotational instability suppressed.The condition of a highly magnetized state over most of the horizon is optimal for the Blandford-Znajek mechanism that generates persistent relativistic jets with \gtrsim 100 efficiency for |a/M|\gtrsim 0.9. A magnetic Rayleigh-Taylor and Kelvin-Helmholtz magnetosphere, which drives a new jet-disk oscillation (JDO) type of quasi-periodic oscillation (QPO) mechanism. The high-frequency QPO has spherical harmonic |m| = 1 mode period of t~ 70GM/c^3 for a/M ~ 0.9 with coherence quality factors Q >/- 10. Overall, our models are qualitatively distinct from most prior MHD simulations (typically, |H/R| << 1 and poloidal flux is limited by initial conditions), so they should prove useful for testing accretion-jet theories and measuring a/M in systems such as SgrA* and M87. Show Partial Abstract