Orange tickets
General
Orange Fri doesn't have fixed pricing, so it not always half price. Early August will differ from late August due to school holidays. Fridays and Sundays are popular for longer trips. Departure time is also a factor.
For Copenhagen - Aarhus tomorrow I find many departures where Orange tickets are sold out, some where the Orange prices are just below the standard ticket, but I also find it close to half price if you go on 5 AM departure.
Copenhagen-Helsingør
For Copenhagen-Helsingør, a CityPass large for 24 hours is usually the best idea, 160 DKK/Adult and 80 DKK/child (12-15 years old). Then you don't have to prebook and it covers local transportation too. But it seems that Orange Fri is always available since a very large majority of passengers don't prebook.
Train pass?
A lot of thing to consider. For example flexibility and youth/children pricing.
Denmark
I believe Denmark is one of the countries where a rail pass could still be a good idea, very few trains require booking and there are alternatives to those. But if you plan to book accommodation ahead of time, you could book some cheap Orange (Fri) tickets between the cities you plan to visit as well. A 17 year old pays adult price and a 14 year old pays child price at 50%.
Sweden
A lot of long-distance trains require seat reservation for €7, while a ticket could be as low as €20. Trains also sells out so the flexibility of a pass is limited. To be safe you need to prebook seats for your pass weeks ahead and then you could just buy the cheap ticket months ahead instead.
SJ is a big operator in Sweden and has a youth discount for you 17-year old similar to the rail pass. A 14-year with adult company only pays 15% of the adult price.
Day trips
In general you won't gain from the rail pass for short day trips. Consider a pass with fewer days instead. But of course, if you need 9 longer trips and choose between a 7 day pass and a 10 day pass, a day trip can make the 10 day pass worth the money.