The young (~16 Myr) pre-main-sequence star in Sco-Cen 1SWASP J140747.93-394542.6, hereafter referred to as J1407, underwent a deep eclipse in 2007 April, bracketed by several shallower eclipses in the surrounding 54d. This has been interpreted as the first detection of an eclipsing ring system circling a substellar object (dubbed J1407b). We report on a search for this companion with Sparse Aperture Mask imaging and direct imaging with both the UT4 VLT and Keck telescopes. Radial velocity measurements of J1407 provide additional constraints on J1407b and on short period companions to the central star. Follow-up photometric monitoring using the Panchromatic Robotic Optical Monitoring and Polarimetry Telescopes (PROMPT)-4 and ROAD observatories during 2012-2014 has not yielded any additional eclipses. Large regions of mass-period space are ruled out for the companion. For circular orbits the companion period is constrained to the range 3.5-13.8 yr (a~2.2-5.6 au), and stellar masses (>80 M_Jup_) are ruled out at 3{sigma} significance over these periods. The complex ring system appears to occupy more than 0.15 of its Hill radius, much larger than its Roche radius and suggesting a ring structure in transition. Further, we demonstrate that the radial velocity of J1407 is consistent with membership in the Upper Cen-Lup subgroup of the Sco-Cen association, and constraints on the rotation period and projected rotational velocity of J1407 are consistent with a stellar inclination of i_*_~68{deg}+/-10{deg}.