It seems like you may be asking a couple of different questions, so the answer depends on what exactly you mean.
First, the two sentences you started with:
When will he come? Tell me.
"When will he come?" implies that somebody is expected to come at some point in the future, and you want to know when he is expected to arrive. If you want to combine that with a request to have somebody tell you that information, then your combined sentence is correct, and means essentially then same thing:
Tell me when he will come.
(Note that the question mark changed to a period, because the main portion of the sentence is "tell me <something>" (a statement, not a question), and "when he will come" becomes a subordinate clause (i.e. the thing that you want to be told to you))
The couple of other forms you listed don't actually mean exactly the same thing:
Tell me when he comes.
This is saying that when he arrives, you want somebody to notify (tell) you at that time (e.g. "Tell me when he comes, so I can come out and meet him.")
Tell me at what time he comes.
This is grammatically correct, but a little more awkward. It could either mean that when he arrives, you want somebody to make a note of what time it is and then you want them to report that time to you at some point in the future, or it could mean "tell me at what time he usually comes", that is, if he often comes at the same time on a regular basis, you want somebody to tell you what that common time is.
Regarding your workplace scenario, if the manager just wants somebody to report to him at some point (maybe at the end of the day, or something) what the recorded time was when the person arrived, then yes "Tell me at what time he comes in today." is the right form for that sort of request. If he wants to be told immediately when the person arrives (so that he can talk to him if he's late, for example), then he would want to say something more like "Tell me when he comes in today." instead.
As for removing the "today", that depends on the context. If it's clear that the person is talking about just today, then there's no real difference, but if they say something like "Tell me what time he comes in" without context, it might be interpreted to mean "keep track of this in general and tell me the times for each day".
As a side-note: In general, it is not very common for people to use the verb "come" by itself. Even though it's not actually grammatically correct, in most cases people will add an appropriate preposition to it even when not using the rest of the prepositional phrase, so for example, if somebody was going to come by (or come over to) your house, you would ask "When will they come by?" or "When will they come over?" instead of "When will they come?". Likewise, in a work scenario, people usually talk about "coming in to work", so the manager would be likely to say "Tell me when he comes in" instead of "Tell me when he comes".
Also, in the case of "Tell me at what time he comes in", that is actually the proper, correct form, but it is very common to leave out the initial preposition ("at"), so it becomes "Tell me what time he comes in". This is actually so common that doing things the correct way (with "at") will often sound a bit strange to people.