Description of a person in chinese
What I understand is that in the sentences 他的个子高高的 the verb 是 is omitted after 个子 in a 是的 construction or no?

他的个子高高的 compare to another similar example, say, 他的肚子肥大.
we say 他的个子高高的 instead of 他的个子高高,
we say 他的肚子肥大 instead of 他的肚子肥大的.
Omitting &quot;是&quot; to construct 是...的 phrase, generally, for some rhetoric reasons, is done only when using twin characters. Like 他的肚子胖胖的，这朵花香香的.
If not with twin words, &quot;是&quot; should not be omitted. We say 他的肚子是肥大的 but never 他的肚子肥大的.
他的肚子肥大 is also right, but 是 and 的 are omitted together.
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For better understanding, imagine this in English:
I got lost in a terrible field. Trees were tall and dense, houses covered with lush mosses, like that everything around was haunted.
&quot;houses covered with lush mosses&quot;, grammatically should be &quot;houses were covered with lush mosses&quot;. But the &quot;were&quot; it omitted. It's for a rhetoric reason, too.

Could you say 他的个子高， without 的

Yes. 的 can be omitted that you can directly say 他个子高.
In Chinese, you can say 他个子高 他个子很高 他头发很少 他肚子胖, with 的 omitted. However, it's informal. Such is just said in casual circumstances, for simplification.

What is the difference between 高高的鼻子 and 鼻子高高的。

There's a difference even in translation:
她有高高的鼻子: He has a high nose
她的鼻子高高的: His nose is high
Just feel the difference in English: He has a high nose -----vs----- His nose is high.

Is it because the clause must have a 有 when using 高高的鼻子 and a 是 may be omitted in the first clause.
For example in the next sentece the 有 could be omitted in many of the clauses.
她有一头卷卷的黄头发，有一双乌黑的葡萄般的大眼睛，有一个高高的鼻子，还有一张樱桃小嘴。”

Yes. It can be omitted so that the sentence is like this: 她有一头卷卷的黄头发，一双乌黑的葡萄般的大眼睛，一个高高的鼻子，还有一张樱桃小嘴 for a rhetoric reason.
Like that you say, for example.
She has a pair of twin-tails, has a couple of big eyes and, has a big long nose.
She has a pair of twin-tails, a couple of big eyes, and a big long nose.
Obviously the second one seems more beautiful.

Another point for me to be clear is that I found this article in Chinese. From what I understand is better not to use 有 when describing people or even may not be right.

NOT not right. It's right. Just seems ugly.
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Notice comparing with the language you've already mastered when learning a new language is important. It can tell you many things.