Is it correct (grammar / style) to shorten the different uses of "ziehen" like this?
The exact answer (in short) is: "It's complicated".
First: it is possible to not repeat (and leave out instead) parts of a sentence if the repeated part would be the same as a part already said. This construction is possible and called Ellipse (ellipsis). There is also a special form of an ellipsis where a Verb is the shortened/contracted piece and this is called Zeugma.
"Ellipse" is (ancient) greek and means "omission". It generally is the omission of parts of a sentence which are implied already and it is used widely, especially in informal speak:
Q: What time is it?
A: seven!
The correct answer would be "It is seven o'clock", but "It is" and "o'clock" are implied.
Notice, this figure of speech can be used and misused as well - with sometimes funny consequences:
Tower: "Flight ABC123, do you have enough fuel or not?"
ABC123: "Yes!"
Tower: "Yes what?"
ABC123: "Yes, Sir!"
"Zeugma" is also a greek word and means "yoke". Two different phrases are joined by a verb which can be used in both but in fact stands there only once:
He took his coat and his vacation.
The sentence works because there is "to take a vaction" and there is "to take a coat" as well, but you may notice that the word "to take" has slightly different meanings here. One could "grab" the coat to take it but "to grab a vacation" would make no sense.
This is why such constructions are often used in a humorous or sarcastic way, playing on the innuendo of different meanings of the word in question:
He went outside, then bonkers.
The same is the case in German:
Ich heiße nicht nur Heinz Erhardt, sondern Sie auch herzlich willkommen.
This classic quote of comedian Heinz Erhardt plays on "ich heiße" (my name is) and "willkommen heißen" (to welcome, verbatim: to name [someone] as [a] welcome[d one]).
On the other hand there are also non-humorous applications of this figure of speech:
Miss Bolo [...] went straight home, in a flood of tears and a sedan-chair.
Charles Dickens, The Pickwick Papers
or, a more classic example:
"Vicit pudorem libido, timorem audacia, rationem amentia."
Desire defeats pudency, audacity fear and folly reason. (Cicero, Pro Cluentio)
Notice, though, that a Zeugma, if used in a non-ironical way, does have to connect two parts via an identical shared part. In English this is easier accomplished but German, with its inflection of nouns, has more room for error. Here is a classic example for a stylistic error:
Was heißt und zu welchem Ende studiert man Universalgeschichte?
"What do we call and to which end do we study Universalhistory?" was the title of the first lecture of Friedrich Schiller at the university of Jena in 1789. The problem with the german sentence is that the continuation of "was heißt" would be a Nominativ (first case) and "wozu studiert" needs an Akkusativ.
Coming back to the picture of a yoke, which is actually a fitting one: you yoke together only two animals of the same kind: oxen are fine and horses are fine, but mixing an ox with a horse is not.