Is walaupun/walau not acceptable in place of namun?

No.
walaupun = although.
namun = however.

yes

I heard "namun" is more formal than "tapi".

Is the repetition of the word "pesan" really necessary in the second clause?

Yes, it is necessary.
'pesan' is the predicate in this clause, and cannot be omitted.
It works like this:

1) Mereka pergi ke sekolah. (S-P-Ket)
2) Mereka ke sekolah. (S-P)
3) Mereka pesan makanan. (S-P-O)

Sentence 1) and 2) have the same meaning but a different syntax.
In sentence 1) the predicate is 'pergi'.
In sentence 2) the predicate is 'ke sekolah'.
In bahasa Indonesia, a sentence does not necessarily need a verb, but it does need a predicate.
The predicate can be something else than a verb.
In sentence 2) the predicate is 'ke sekolah' ==> adverbial complement.
In sentence 3) the predicate is 'pesan' ==> verb.
The verb 'pesan' is used here as a transitive verb with 'makanan' as the object.
If you leave out 'pesan', then you would leave out the predicate, and the sentence will be grammatically incorrect.
You would end up with Subject-Object (S-O) without a verb, and that doesn't work.
Subject-Predicate (S-P) is the minimum requirement for an Indonesian sentence.

Here is more info about the sentence structure:
https://forum.duolingo.com/comment/29750700

The good news is that you can leave out other elements.
Examples:

4) Mereka pesan kopi, namun mereka tidak pesan makanan.
5) Mereka pesan kopi. Namun mereka tidak pesan makanan.
6) Mereka pesan kopi, tapi mereka tidak pesan makanan.
7) Mereka pesan kopi, tapi tidak pesan makanan.
8) Meraka pesan kopi, bukan makanan.

In the formal style ('baku'), sentence 4) should actually not be allowed.
It should be like sentence 5) or 6).
'Namun' is used to connect two sentences, it can only appear after a period, not after a comma.
'tapi'/'tetapi' is used to connect two clauses and can only appear after a comma.

If we use sentence 6) as a new example, then you could leave out 'mereka' in the second part.
6) Mereka pesan kopi, tapi mereka tidak pesan makanan.
The subject of the first and the second clause is the same and you can leave out the second subject.
That will give you sentence 7)
You could also rephrase the second clause and transform it into sentence 8) with only one clause.

Please also note that sentence 8) uses another negation word.
'bukan' is used to negate the noun 'makanan'.
'tidak' is used to negate the verb 'pesan' in the other sentences.

Here is more info about negation words:
https://forum.duolingo.com/comment/31291121

Let's have a look at the other sentence that you mentioned :
"Mereka pergi dari sini, namun mereka tidak ke sekolah."
I have the same problem with this sentence.
To be grammatically correct in the formal style, the comma should be replaced by a period.
Another possibility is to replace 'namun' with 'tapi' / 'tetapi'.
'namun' is to connect two sentences, not two clauses.
(see difference between sentences 4-5 and 9-10).
(see KBBI definitions below).
Examples:

9) Mereka pergi dari sini, namun mereka tidak ke sekolah.
10) Mereka pergi dari sini. Namun mereka tidak ke sekolah.
11) Mereka pergi dari sini, tapi mereka tidak pergi ke sekolah.
12) Mereka pergi dari sini, tapi mereka tidak ke sekolah.
13) Mereka pergi dari sini, tapi tidak ke sekolah.

The difference between the example sentences is only in the second clause.
Let's use sentence 11) as the new base sentence.
The second clause of sentence 11) and 12) is the negation of sentence 1) and 2).
Sentence 13) leaves out the subject of the second clause (just like in sentence 7).

To complete the picture, here are the KBBI definitions of 'namun' and 'tetapi' :

https://kbbi.web.id/namun

namun/na·mun/ p
kata penghubung antarkalimat untuk menandai perlawanan; tetapi

As you can see, 'tetapi' is given as a synonym.
It has the same meaning but a different usage.
(see next definition)

https://kbbi.web.id/tetapi

tetapi/te·ta·pi/ p
kata penghubung intrakalimat untuk menyatakan hal yang bertentangan atau tidak selaras

These two words are used differently:
namun ==> antarkalimat = between the sentences (connecting two sentences).
tetapi ==> intrakalimat = in the sentence (connecting two clauses).