Copenhagen - Hamburg has mandatory seat reservations in summer. Check if you can book it here https://travel.b-europe.com/dsb-rail/en/reservation-only#TravelWish
Maybe it is not available yet.
Milan - Barcelona would be best done on 2 travel days. That way you don’t have to pay expensive seat reservations or even pay for the Frecciarossa train to Chambéry. Keep in mind that the Italian summer timetable hasn’t been entirely published yet.
Itinerary (as an example) :
DAY 1
- IC Milano Centrale - Ventimiglia 07:10 - 10:54 3€
- TER Ventimiglia - Nice 11:16 - 12:11
- TER Nice - Marseille 13:25 - 16:03
- TER Marseille - Avignon (stay for the night in Avignon) 16:55 - 18:08
DAY 2
- TER Avignon Centre - Portbou 09:31 - 13:47
- Rodalies Portbou - Barcelona Passeig de Gracia 14:35 - 17:04
Of course, it is longer than TGV rides but it saves you a lot of money.
Madrid - Lisbon requires a full day of travel. See :
https://www.seat61.com/trains-and-routes/madrid-to-lisbon-by-train.htm
There are problems with the direct trains between Hamburg and Copenhagen during the summer as the plan was to start using new trains on the route. However, it is still unclear when these trains will be available and therefore reservations can't be made at the moment.
Copenhagen - Hamburg has mandatory seat reservations in summer. Check if you can book it here https://travel.b-europe.com/dsb-rail/en/reservation-only#TravelWish
Maybe it is not available yet.
@thibcabe for the moment no news about a mandatory reservation on summer on Hamburg-Copenhagen. Like Anna said, soon should DSB start using old DB IC carriages for this route with more capacity, but some documentation for using the Big Belt bridge are missing. For the moment we don't know if there will be a compulsory reservation and if there will run the "new" trains.