After moderate training, the same day is fine and often ideal. After maximal efforts or competition, wait about 24 hours to let the acute inflammation settle. The massage will feel better and accomplish more.
After a light or moderate session, yes. After deep tissue work, give the tissue 24 to 48 hours before your next heavy load; light movement during that window actually helps.
No. This concern circulates in gyms, but current evidence shows no negative effect on hypertrophy. Indirectly the effect is positive: better recovery means more consistent training, and consistency is what builds muscle.
After, for nearly every goal. Reserve pre-exercise work for short, light preparation, and keep anything deep at least a full day away from heavy sessions and competition.