Issuing a command immediately
Your first question:
Can the caster command a first order to the dominated creature (e.g., 'move closer to me') immediately, or must the command be issued on the caster's next turn?
As you point out, the spell says "no action required".
Furthermore, the "Other Activity on Your Turn" in the Combat Chapter of the Basic Rules says:
You can communicate however you are able, through brief utterances and gestures, as you take your turn.
So, allowing the caster to issue a single command immediately seems pretty reasonable.
However, the "no action required" phrase could conceivably require some DM interpretation. I don't think it's granting you a super-power, it just means issuing commands via the telepathic link doesn't take an action, especially contrasted to taking total control. It's reasonable that commands via the telepathic link would be adjudicated the same way verbal communication is. The rules certainly don't suggest you can talk verbally on other people's turns.
When does the dominated creature act?
Your second question:
Related to the previous question: does the dominated creature follow that order on its initiative?
Yes. Nothing suggests anything different.
When under total control, when does the dominated creature act?
Your third question:
When the caster uses their action to take total control, does the creature act at the same time as the caster or on its own turn?
Its own turn. Again, nothing suggests anything different.
Finally
Obviously, the DM can deviate from this interpretation, and probably should under some circumstances. For instance, often DMs have controlled creatures take their turns immediately after the controlling player if only as part of executing combat efficiently. As another example, if the target takes its turn, then the caster dominates via the spell, it's an entire round of combat until the dominated creature's turn comes up. In such a case the DM might allow the caster to revise or reissue a telepathic command.
