Trains in Denmark

I think you will either purchase the Stockholm-Lund tickets on Swedish Rail
or their app. I have had trouble getting Swedish trains to show up on trainline
in the past. Also, I believe there are multiple apps that can potentially show you
the same info, so be careful not to buy the same ticket more than once.

The Danish website is DSB

I am pretty sure the Swedish trains can go up in price and the Danish ones may
also, so if you are certain of your travel times, purchasing ahead is the way to go.
If you are not sure, then you can purchase on the app right before you board
or at a machine in the station.

Do you know the name of the app? I could also look it up. :)

I'm a bit confused about your plans; "Lund to Aarhus" vs. "Aarhus to Koge (Køge?), then Koge to Copenhagen". How did Køge get into this???

But here are some thoughts:

just purchased the tickets when we get to the station

You can absolutely do this, but I don't recommend it. The train will not be sold out, but it might be difficult to get seats together. And you cannot get discounted "Orange" tickets. Orange tickets come with compulsory seat reservations, but I would always get that anyway. My legs and back are not up to standing the whole way, so I want to be sure of my seat. You do you.

The link above to DSB is fine, but remember that the stations will pop up with the Danish names (København H = Copenhagen Central Station; Aarhus H = Aarhus Central Station)

In my experience of having used InterCity and Lyn trains in Denmark relatively often in recent years: If you are two people, there is a very high chance of getting seats together even when booking them just a few hours before departure, but exceptions to this does occur sometimes (and for me those times seem kind of random). If you are more than two people, it can be a good idea to book seats earlier if you want to be sure to sit together.

The "Standard" prices you see on DSB are fixed fares, and is the same as you'd pay on the travel day. But as you can see on their website and app, you can save a good amount of money if you book early with one of the "Orange" or "Orange Fri" tickets, which offer no/limited flexibility.

And a small correction to the reply above: Orange tickets do not come with compulsory seat reservations anymore. That was changed quite a few years ago.

Orange tickets do not come with compulsory seat reservations anymore

Thank you :-)