"It does not matter where he comes from."
Translation:Es ist egal, woher er kommt.
"Es ist egal, woher er kommt." Here, "woher er kommt / where he comes from" is a relative clause. The main clause is "Es ist egal", and it needs a verb to be complete.
"Egal, woher er kommt, er ist willkommen." = "No matter where he comes from: he is welcome." Same as in English, you don't need to say "es ist egal / it doesn't matter" here.
Is saying "Es ist egal, wo er aus kommt." totally wrong in German?
Yes, totally wrong.
The closest possible sentences to that are:
1 - Es ist egal, woraus er kommt. (Prepositions come before the thing they modify – but aus was turns into woraus.)
This means "It doesn't matter what he comes out of." and implies that he's inside a small container. We wouldn't use this for talking about a city or country of origin.
2 - Es ist egal, wo er auskommt. (Separable prefixes come at the end of a clause, but if the verb itself also goes to the end, it joins up with the separable prefix.)
This means "It doesn't matter where he makes do." (from auskommen "manage, make do, get along, get by")
3 - Es ist egal, wo er herauskommt.
This means "It doesn't matter where he comes out. / It doesn't matter which exit he uses."
None of them mean "It doesn't matter where he comes from."
And ..., wo er aus kommt. is grammatically impossible.
woher here starts a relative clause.
Relative clauses in German are subordinate clauses and so the conjugated verb - here, kommt - has to be at the end.
Thus you have woher ........ kommt, and since er is the only other word in the clause there's only one place for it to go.
As stepintime explains above, just "egal" is equivalent to "no matter," not "it doesn't matter." You need "Es ist egal" to express the latter: "It is no matter" -> "It doesn't matter."
So "Egal, woher er kommt" would just mean "No matter where he comes from" and would need another clause after it to make sense.
"It doesn't matter where he comes from" is a full sentence. "No matter where he comes from" is a fragment and needs another clause to complete the meaning-- e.g., "No matter where he comes from, he is welcome here." You can't just say "No matter where he comes from"; that's not a complete idea and wouldn't make any sense.
So, why is this other example from Duolingo valid?
"Egal, wie teuer es ist."
By the same rule, it would need a second clause, wouldn't it?
"Wo" by itself specifies a location no matter what else is in the sentence (i.e., at some place rather than from some place). You need "woher" to show the movement of coming from someplace (and for going to someplace, you would need "wohin").
"Auskommen" doesn't mean coming from somewhere; just use "kommen." Also, use the "-t" ending since the subject is "er": "kommt."
So "Es ist egal, woher er kommt."
I'm guessing that es zähnt nicht is an error for es zählt nicht but that means "it doesn't count" rather than "it doesn't matter".
That might work if there were more context, e.g. Es zählt nicht, woher er kommt, sondern vielmehr, was er kann "What counts is not where he comes from but rather what he can do".