Does time travel by any member of the Extraterrestrial Alliance sometimes conflict with what other member races have set in motion to influence something or someone on the Earth, or aims of the Alliance itself? If so, how are such conflicting interests managed?

If you consider the implications here, that one decision to do time travel affects everyone galaxywide, each of the civilizations is in a position to be manipulating the others and everything else in the bargain. So this has become the privilege of the proficient, that an agreement exists to coordinate among the parties when time travel is desired for any reason. This helps keep the peace and is a necessary consideration.