Does run count

[3ump]

VFW ball major league rules: Runner on first ball hit to centerfield, fielder makes good catch at shoe tops and comes up throwing to get runner before return to first . Ball goes out of play and runner (a) touches back on first or (b) seeing ball go out of play goes onto third.

Second part of question: If scenario (b) once ball is put back in play can defense appeal that runner never tug up before advancing and would appeal be upheld.