Learn Russian: С одной стороны, он прав, с другой стороны, так нельзя. - On the one hand he is right, on the other hand one cannot act like that.

In Russian, this idiom literally translates as "from one side". The preposition с meaning "from" requires the genitive case, which is why the feminine noun сторона changes to стороны.

Is it mandatory to repeat стороны in the second part of the sentence?

No, you can safely drop it. Just as in English you can say "on the one hand... on the other", in Russian it is very common and natural to just say с другой and leave out the second стороны.

Why do we use the short-form adjective прав instead of the full form правый?

We use the short form прав because it describes a state or an assessment of a specific situation ("he is right in this instance"). The long form правый is generally used for permanent attributes, directional meanings ("the right side"), or to modify a noun directly.

The translation says "one cannot act like that", but the Russian phrase так нельзя has no verb or subject. How does this work?

Нельзя is an impersonal word meaning "it is impossible" or "it is not allowed". In Russian, you don't need a dummy subject like "it" or "one", nor a linking verb like "is" in the present tense. The phrase так нельзя is a very common standalone expression meaning "you can't do that" or "that's not okay".