(1) Some discounted tickets marketed as "DSB Orange" are only valid on a specific train, but ordinary tickets will be valid on any train no matter whether you have a matching seat reservation.
If you don't catch the train you have a reservation for, you will of course not be guaranteed a seat, but unless you run into the outbound commuter peak, your chances of finding a place to sit are excellent.
Most Danish trains don't require seat reservations; the exceptions are some InterCity Bornholm departures, which are not relevant for you.
(2) The wifi in intercity trains is free. There is a mandatory sign-on step, but it does not require a phone or even any credentials:
- Sign on to the wifi network named "TDC".
- Try to load any HTTP (not HTTPS) page; you will be redirected to a capture page in Danish.
- Ignore all the text, and click the big red button marked "Klik her".
- You now have internet connection.
(This answer typed on the train after trying out above procedure)