We analyze 221 eclipsing binaries (EBs) in the Large Magellanic Cloud with B-type main-sequence (MS) primaries ($M_1$ $approx$ 4 - 14 M$_{odot}$) and orbital periods $P$ = 20 - 50 days that were photometrically monitored by the Optical Gravitational Lensing Experiment. We utilize our three-stage automated pipeline to (1) classify all 221 EBs, (2) fit physical models to the light curves of 130 detached well-defined EBs from which unique parameters can be determined, and (3) recover the intrinsic binary statistics by correcting for selection effects. We uncover two statistically significant trends with age. First, younger EBs tend to reside in dustier environments with larger photometric extinctions, an empirical relation that can be implemented when modeling stellar populations. Second, younger EBs generally have large eccentricities. This demonstrates that massive binaries at moderate orbital periods are born with a Maxwellian thermal orbital velocity distribution, which indicates they formed via dynamical interactions. In addition, the age-eccentricity anticorrelation provides a direct constraint for tidal evolution in highly eccentric binaries containing hot MS stars with radiative envelopes. The intrinsic fraction of B-type MS stars with stellar companions $q$ = $M_2$/$M_1$ $>$ 0.2 and orbital periods $P$ = 20 - 50 days is (7 $pm$ 2)%. We find early-type binaries at $P$ = 20 - 50 days are weighted significantly toward small mass ratios $q$ $approx$ 0.2 - 0.3, which is different than the results from previous observations of closer binaries with $P$ $<$ 20 days. This indicates that early-type binaries at slightly wider orbital separations have experienced substantially less coevolution and competitive accretion during their formation in the circumbinary disk.
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