to quote myself from an other discussion about Denmark:

"Not sure how it exactly works, but I tried to buy (a pretty standard) 3 sim card last December. They didn't want to sell it to me without some Danish ID card. So it is possible that you need a Danish ID card or at least the CPR number to buy a prepaid simcard. Which would be pretty stupid, but still."

So not really sure if you will be able to get a Danish sim card.

Data on national roaming (on TIM) is not available for free also for Italian customers of 3 ITA, so it doesn't change anything for us. The 3 like home is really "like home", but for data only, for calls you pay a set-up fee.