When do couples hold hands in Korea? Is there an order to skinship?

It splits before and after the confession. There is no set number of dates; what matters is whether the relationship has been made official.
Before the confession, taking someone's hand or putting an arm around their shoulder reads as a signal of interest. It works as something said without words.
After the confession, holding hands and linking arms become ordinary. This is why so many couples on Seoul streets are walking arm in arm.
The word skinship is itself a Korean coinage from English. It does not mean anything sexual. It covers closeness in general: holding hands, linking arms, leaning on someone.
In public, hand-holding and a light hug are the usual limit. Kissing is uncommon in the open, and people are especially careful in front of older generations.
The same applies at concerts. Queues and seated rows are packed spaces, so anything more than that stands out.